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As far as I understand, all banks can borrow money from the Central Bank (Federal Reserve in the US) and the rates are (currently) extremely low (or even negative in some cases). If that is so, why would a bank need to accept deposits from private persons? While some banks do charge fees for keeping deposits, there are also plenty of free-of-charge offers out there.
In order not to make this too broad, assume that the bank is in the US.
They don't actually need to. They accept deposits for historical reasons and because they make money doing so, but there's nothing key to their business that requires them to do so.
Here's a decent summary, but I'll explain in great detail below.
By making loans, banks create money. This is what we mean when we say the monetary supply is endogenous. (At least if you believe Sir Mervyn King, who used to run England's central bank...) The only real checks on this are regulatory--capitalization requirements and reserve requirements, which impose a sort of tax on a bank's circulating loans. I'll get into that later.
Let's start with
Why should you believe that story--that loans create deposits? It seems like a bizarre assertion. But it actually matches how banks behave in practice. If you go borrow money from a bank, the loan officer will do many things. She'll want to look at your credit history. She'll want to look at your income and assets. She'll want to look at what kind of collateral or guarantees you're providing that the loan will be repaid. What she will not do is call down to the vaults and make sure that there's enough bills stacked up for them to lend out. Loans are judged based on a profitability function determined by the interest rate and the loan risk. If those add up to "profitable", the bank makes the loan. So the limiting factor on the loans a bank makes are the available creditworthy borrowers--not the bank's stock of cash.
Further, the story makes sense because loans are how banks make money. If a bank that was short of money suddenly stopped making loans, it'd be screwed: no new loans = no way to make money to pay back depositors and also keep the lights on = no more bank.
And the story is believable because of the way banks make so little effort to solicit commercial deposit business. Oh sure, they used to give you a free toaster if you opened an account; but now it's really quite challenging to find a no-fee checking account that doesn't impose a super-high deposit limit. And the interest paid on savings deposits is asymptotically approaching zero. If banks actually needed your deposits, they'd be making a lot more of effort to get them. I mean, they won't turn up their noses; your deposited allowance is a couple basis points cheaper to the bank than borrowing from the Fed; but banks seem to value small-potatoes depositors more as a source of fees and sales opportunities for services and consumer credit than as a source of cash. (It's a bit different if you get north of seven figures, but smaller depositors aren't really worth the hassle just for their cash.)
This is where someone will mention the regulatory requirements of fractional reserve banking: banks are obliged by regulators to keep enough cash on hand to pay out a certain percentage of deposits. Note nothing about loans was said in that statement: this requirement does not serve as a check on the bank making bad loans, because the bank is ultimately liable to all its depositors for the full value of their deposits; it's more making sure they have enough liquidity to prevent bank runs, the self-fulfilling prophecy in which an undercapitalized bank could be forced into bankruptcy. As you noted in your question, banks can always borrow from the Fed at the Fed Discount Rate (or from other banks at the interbank overnight rate, which is a little lower) to meet this requirement. They do have to pledge collateral, but loans themselves are collateral, so this doesn't present much of a problem.
In terms of paying off depositors if the bank should collapse (and minimizing the amount of FDIC insurance payout from the government), it's really capital requirements that are actually important. I.E. the bank has to have investors who don't have a right to be paid back and whose investment is on the hook if the bank goes belly-up. But that's just a safeguard for the depositors; it doesn't really have anything to do with loans other than that bad loans are the main reason a bank might go under.
Banks, like any other private business, have assets (things of value) and liabilities (obligations to other people). But banking assets and liabilities are counterintuitive. The bank's assets are loans, because they are theoretically recoverable (the principal) and also generate a revenue stream (the interest payments). The money the bank holds in deposits is actually a liability, because it has to pay that money out to depositors on demand, and the deposited money will never (by itself) bring the bank any revenue at all. In fact, it's a drain, because the bank needs to pay interest to its depositors. (Well, they used to anyway.)
So what happens when a bank makes a loan? From a balance sheet perspective, strangely enough, the answer is nothing at all. If I grant you a loan, the minute we shake hands and you sign the paperwork, a teller types on a keyboard and money appears in your account. Your account with my bank. My bank has simultaneously created an asset (the loan you now have to repay me) and an equal-sized liability (the funds I loaned you, which are now deposited in your account). I'll make money on the deal, because the interest you owe me is a much higher rate than the interest I pay on your deposits, or the rate I'd have to pay if I need to borrow cash to cover your withdrawal. (I might just have the cash on hand anyway from interest and origination fees and whatnot from previous loans.)
From an accounting perspective, nothing has happened to my balance sheet, but suddenly you owe me closing costs and a stream of extraneous interest payments. (Nice work if you can get it...)
Okay, so I've exhaustively demonstrated that I don't need to take deposits to make loans. But we live in a world where banks do! Here's a few reasons:
You can probably think of more, but at the end of the day, a bank should be designed so that if every single (non-borrowing) depositor withdrew their deposits, the bank wouldn't collapse or cease to exist.
borrow money from the Central Bank
Wrong premise. They cannot borrow as much as they want and they cannot borrow without collateral i.e. government debt instruments they hold or any other instrument with value. And banks don’t have unlimited collateral to borrow against.
Secondly central banks aren’t in the business of lending unlimited money. The more money they lend out, the more is the money supply which stokes inflation which will eventually lead them to stop lending. At any point of time they want a certain amount of money movement, so they can control inflation and interest rates within an agreed limit and as limited by their economy. No sane central bank would want to stoke hyperflation by printing money at will e,g, Helicopter money. So the only other way for banks is to accept deposits from private individuals.
You can also argue that banks make money by connecting lenders and borrowers and make their profit by being the middleman without using their assets. So you can say they are making a profit with the minimum usage of their capital. Albeit they have the central bank looking over their shoulder to police their behaviour.
While some banks do charge fees for keeping deposits
Yes but many provide certain extra services for which they charge. That is how they differentiate between no fee accounts and fee paying accounts.
Banks cannot just borrow from the Federal Reserve and use that money to make loans. The first thing you need to understand is how fractional reserve banking works. The banks can make loans with money that their customers have deposited in their accounts. The interest and fees from those loans go to pay the salaries of those working at the banks with leftover profit to pay dividends (interest on your bank accounts).
The only reason that the Federal Reserve allows overnight lending is so that banks don't immediately become insolvent if they have larger than usual withdrawals by their depositors. The Federal Reserve keeps an eye on the balance sheets of the banks that are doing the borrowing, and if they didn't have assets in the form of deposits, they would force the banks to sell the loans that were made from those deposits.
What does this have to do with personal finance? I think this question is only marginally on-topic here. This amount of money in circulation is affected specifically by the fraction of the money that can be used for making other loans. But the bigger influence is the rate that the Federal Reserve charges for overnight lending. They raise and lower the rates which affects the rates that the banks can lend at while remaining profitable.
Borrowing money from the Federal Reserve (or other central banks) requires full collateral, generally in terms of treasury bonds. In that sense it is only a source of liquidity - getting short term money by pledging guaranteed future cash flows, not random commercial loans. To get a dollar from FR today requires freezing a dollar that you already had.
Private deposits, on the other hand, require only a keeping a fraction of them as reserves, so you can use the rest of the money for new loans.
They don't need to accept deposits from normal persons, but that's how they make lots of money. Banks make money off the fees they charge retailers when those folks swipe their debit cards at the retailer. It's their bread and butter. In order to facilitate you accruing swipe fees for them, they need to allow you to make deposits, on which they can charge the retailers swipe fees.
Let's just focus on the "why would a bank need to accept deposits from private clients part" and forget the central bank for a moment.
I'm a guy. I have a wife and two kids. They have this pesky habit of wanting to buy stuff. When I get paid, I could just get a check, cash it, stuff it under a mattress, and pull it out when I need it. Hey that worked for a long time didn't it?
But sometimes it's nice to write checks. (Just kidding, that's so gauche...) I use my debit card. I use my credit cards, but they need to be paid somehow. My light and phone bills need to be paid too. If only there were someone out there who could facilitate this transfer of money between me, the private client and the merchants I'm forced to spend my money at.
Now some of those merchants have plans. Light bills I can pay at my grocer if I choose. But most of the other's don't. Luckily I have a bank that's willing to do this, for a fee. So basically they do it because there's a void in the market if they don't. I don't know if it's true what they say about supply creating its own demand, but it certainly is true that demand creates supply!
Central Banks are essentially a cartel, designed to let banks in general borrow money from depositors at relatively low interest rates. They do this in two ways: By reassuring depositors that momentary cash flow problems at banks will not result in banks failing, they lower the interest rates that depositors demand. And by imposing strict regulations on banks that are borrowing from depositors at high interest rates. (People who move money to the banks offering the highest interest rates are especially likely to participate in bank runs.)
Borrowing "too much" from the Central Bank is considered to be a sign of a bank that is too weak to attract deposits from depositors at "reasonable" interest rates.
If a bank borrows "too much" (as a percentage of the bank's assets) from the Central Bank, the bank regulators will subject the bank to heavy scrutiny. If the bank fails to find ways to reduce its borrowing from the Central Bank, the bank regulators are likely to steal the bank from its shareholders, and sell the bank to a "stronger" bank that pays lower interest rates.
