Having a social security or other tax-related ID has sufficed for banks so far, which doesn't guarantee the holder is a citizen but does demonstrate enough relevant "status" with the government for banking to probably go smoothly.
Digging ourselves deeper into our already awful decentralized partially-privatized (the CRAs, mostly) identification system by expanding the set of things we have to prove in even more circumstances is not a good thing.
In most countries of the world, the best way to prove your citizenship is to apply for a visa. That is you world apply for a US visa and get an official rejection, because US citizens don't need/cannot get a visa, and the rejection document would be the proof of citizenship.
Or a FS-240, Consular Report of Birth Abroad, from the State Department. I was born on a US military base and although I have a birth certificate, the only think I've ever been able to use it for was my REAL ID. I had to use the FS-240 for my passport, SSN, etc.