Have you looked into a local credit union?
I have my checking and savings accounts linked- the checking account can be at $0.00 and any check that comes through then gets sucked out of the savings account. No fee for that.
They also give you dividends, but it's not much to write home about.
I have all the online access that I need- billpay, transfers, etc.
ATM use is usually free at other CUs.

I'm afraid I don't see the attraction to web-based banking the way you've described it.