What? It's always worked for me.
I'd say that USAA is the absolute best in terms of their web and app, but I'm not sure why you can't deposit checks.
My second favorite is Bank of America.
Same here (second generation member, non-military). I don't think I've ever actually tried to use the app-based MobileDeposit, but it works like a champ via the website (plus scanner).
I love the USAA app. I wish the budgeting tools were a bit more robust, as I could probably drop my monthly budget app and go 100% USAA. But the ability to manage accounts (USAA and non-USAA), transfer between accounts, pay bills, etc. all from the phone are incredibly simple.
I'm excited to see what they do for iOS 7.