Just an online account is usually just fine
I have had accounts only at online banks for several years (Schwab Bank if interested). Before that I had an account at Wells Fargo myself but that is now closed due to poor service and high fees I ended up paying. (This answer is based on my own experiences but should apply equally to anyone in a similar circumstance.)
Most services are available remotely
The bank sends me checks, pre-paid envelopes to mail in deposits or other written correspondence, a debit card (which is free to use at any ATM worldwide, fees are refunded) and I can deposit most checks over the phone, with very large checks needing to be mailed in.
The only caveat as you say is depositing cash. This is significantly more awkward. Personally when I have received cash, I stockpile it for a few months, then buy a money-gram (available at most large grocery stores), or cashiers-check written out to myself, and then deposit it as usual. This does cost a few cents to a dollar or two depending where you go, but at least for myself, this is greatly preferable to an account which charges fees, which all banks with a significant physical presence seem to do.
Overall I would say most consumers could do fine with only an online checking account and no accounts from a physical bank, unless you very frequently handle cash (such as having a cash / tip paying job). If you do it is likely worth having a bank account at a physical bank simply to avoid the fees that come with cash handling.
Do you actually want to invest?
This does not directly answer the question but I would add as an aside, even if you find an account with exceptionally good interest rates, if you plan to keep your money there for any length of time, it may be worth instead investing it. Even with very safe investments one can comfortably earn 5% a year, as opposed to around 2% from the best bank account rates. I wont go into details here as it was not what was asked, but you should consider platforms other than banks if you want money to grow for you.
(The one caveat I will add, is Schwab Bank does actually have a few branches, which one can use in case of emergency, but are usually hundreds of miles away. I have had to go to one once ever. They do not offer normal bank services but are only for exceptional use.)