They would have two options:

A) keep the euro notes in their vault, which only works if you deposit paper bills in the first place

B) keep electronic deposits in the ECB and pay interests to do so

In either case if they give back the money to the clients when they ask for it how do you expect them to cover their operating costs (plus the interest they are charged by the central bank in case b)?

You mean like... a German bank? They will happily do what you ask. For a small fee. In the form of a negative interest rate.

And your deposit won't be insured past like 100k euro.

This imaginary bank would still have costs. What would be their income?

Excellent question. There is a company, The Narrow Bank, that has the same idea, but they didn’t get a banking license from the Fed. Matt Levine, whose newsletter you should clearly start reading, has the details: https://www.bloomberg.com/opinion/articles/2018-09-06/fed-re...

This is the first responsive answer I've gotten to that question.