Yea again this assumes your demo code proves to be correct on the first swing, and I can count on one hand I've seen that happen.
I honestly don't see the point in writing tests for demo/poc code until you and the client are in agreement that the current stage looks correct. Changes after that are changes in scope and will probably require tests to be reworked anyway.
How do you know your top level API tests are stable? How do you know they're going to stay that way?
Three thousand years of beautiful tradition, from Moses to Sandy Koufax, you're god damn right I'm living in the fucking past