You're right, in this model it wouldn't be nearly as profitable to be a bank as it is now. Potentially, it could be a "public good" provided by the government, or it could be like a lot of the brokerages who provide investment services and charge a commission on top.
Also, you're right it would be very hard for banks to be anywhere near as profitable as they are now if assets are backed 1:1.
But in some sense, banking is already viewed as a public good since the FDIC is backing all of the bank deposits. So at the moment we kind of have a weird hybrid situation where banks are kind of pseudo-private, but also backed by the government.