On Schwab, I have both a checking and a brokerage account. When I want to buy stock/ETFs, I have to transfer money from one to the other. I can just do this from a webpage, relatively instantly. What's the purpose (for the bank—for me it seems to just add overhead) of having them separated, rather than just letting me buy stock from my checking account funds?
It's because investment accounts are legally required to be separate from deposit accounts, and the bank must manage them separately as well.
Deposit accounts (i.e checking and savings) are only for cash and cash equivalents. The money you put into a deposit account is (in spirit) merely being held by your bank for safety and convenience purposes, and they must give it back to you any time you ask for it. Therefore, banks are legally limited in what they can do with it while you're entrusting it to them. Yes, the bank does lend that money to other depositors, putting it at some risk, but they are heavily regulated on how they do that. Ideally, since you can never lose your deposit, the bank isn't allowed to lose it either. Deposit accounts are guaranteed (up to $250,000) by the FDIC, which has regulatory power over commercial deposit banks and can preemptively shut down or otherwise intervene to ensure banks can make good on their deposits.
Investment (brokerage) accounts, on the other hand, are for the purpose of purchasing (possibly risky) investments that can potentially lose money. Any investments you make, you're on your own. The bank is only required to prohibit you from doing a few specific things thay are very risky and/or stupid. The "cash" portion of your investment account is money that the bank can use to purchase it's own (possibly risky) investments. Although they have a fiduciary requirement to protect your money, this is not the same thing as a guarantee. Investment banks are insured by the SIPC, but unlike the FDIC, it has no regulatory power and does not insure the actual value of your investments.
In short, banks have much more flexibility with investment money than they do with deposit money. This is why you have to do a transfer before you can invest.
The reason things are this way is because of some very shady activities that banks were engaging in during the Great Depression, and during the subprime mortgage crisis.
I think the issue is fraud exposure and budgeting. Schwab actually gives you the option to do it either way.
As you know, many brokerages offer a service where you can write checks, use a debit card, pay bills, and buy stocks against the cash in a brokerage account. The cash is held at a FDIC-insured bank or automatically swept between money market funds. This is known as a cash management account and eliminates the need for manual transfers between investments and daily spending.
Schwab has a list of options. Looking at the list you can see that you can indeed use a debit card and write checks against uninvested cash. That cash can be swept into FDIC insured bank accounts. You also have "Schwab Bank High Yield Investor Checking Account" which is nearly exactly the same, except you have to manually transfer the money.
I believe they make a distinction because I have heard complaints that people don't want to have large amounts of cash at risk of fraudulent debit card transactions or checks. For example, somebody may be uncomfortable using a debit card at a online store that has $30,000 behind it to buy something small. Or somebody could add an extra zero to a check and it would clear.
By keeping a relatively small balance in the checking portion that they can manually control, they will not worry as much.
Additionally, having a separate balance does make budgeting easier by allowing you to see your monthly spend and budget. Investment accounts tend to fluctuate, as you buy and sell stocks and receive dividends. Is the large drop in your cash because you spent too much this month or merely bought stocks? Was the increase in your portfolio due to the markets going up, or did you forget to pay your mortgage this month?
Having a separate balance lets you segregate your long-term savings from your monthly finances.
It doesn't necessarily need to be separate. For example: Interactive Brokers offer direct deposit, debit card and bill payment directly from one's brokerage account.
The brokerage account would fall under SIPC protections if something were to go awry (as opposed to FIDC insurance that applies to bank accounts).