Finally seems I got it.

Driver's license is issued by states, and they aaccounted as valid ID for voting. There is no way to force common fed rules on the internal business of state in issuig driver license.

So if some state will not demand birth certificate for driver's license, then there is no way to cut non-citizens from voting - they will show that state driver's license and should be allowed to vote as citizen.

Birth certificate have no photo, so it can't be used as proof of citizenship at voting stations.

Then I don't see any solution other than demand birth certificate along with driver's license at voting stations or implement a new federal (or issued by state under strict fed rules) document. The latter then will give feds unnecessary additional power over all US, at least in question of voting.

Democrats resist voting ID because they need illegal votes of non-citizens, and at the same time conservatives could view voter ID as a feds overreach.

Why not demand birth certificate or whatever document foreigners get, when granted legal citizenship, as a proof of citizenship, in addition to any document with photo at voting stations? No additional suspicious fed IDs, no way for non-citizens to vote (at least massively).