I have chase, USAA (which is basically an online bank since there aren't any branches) and also American Express has a savings online bank as well.
I've only gone into a physical bank 2 times in the past 5 years, but it was when I was in San Antonio and needed a giant cashier's check from USAA. Everything else you never need to go into including bill pay, checks, free ATM fee's, deposits ect... Almost all deposits I do with my phone (take a picture of the check and boom it's there).
I keep Chase because they have so many ATM's around the United States that you can use one for cash deposits. Literally the only reason I have an account.
Go for it.