You can do that with a check but you can't really take smartphone pictures or use mail to deposit cash. I have an online bank account but I do maintain a checking account with a traditional US bank for this reason.
This is true, and my bank does have one branch in my city I can go to if I ever had to deposit cash. In actuality, though, I rarely receive large sums of cash and any cash I do get I usually just spend (thus avoiding having even to visit an atm to get more cash as I commonly have to). The end result is that I haven't been inside a bank in at least 5 years.
