Do you have some options that have no strings (like minimum balance) attached?
Instead, try local credit unions. As an example where I'm at, Indiana Members Credit Union has a checking account with all the normal perks of a checking account and no fees. And credit unions work together with the major banks to give their members access at most ATMs.
In any capital one branch you can make a deposit/withdrawal with your 360-checking account. If they give you freedom of online-banking it doesn't mean their branches magically become unaccessible to you.