You hit the nail on the head. They're viewed (and valued) as an app on top of a bank. Without holding and investing deposits, they don't get the other financial benefits. Perhaps their end user data could be worth something, though.
TL;DR Its about the transaction volume.
They have a lot of costs for that interchange fee though. That's not where banks make they money and the .01% interest you mention is what they pay, not what they would earn if they actually held your deposits. If they held your deposits and worked like a regular bank it would be dramatically more profitable: $10,000 in your account means they would owe you $1 a year in interest. They could simply park that about anywhere else and make a decent amount, but banks really make their money by leveraging.