It's a solution without a problem which is almost certain to cause massive problems and delays, a huge number of which will almost certainly affect people who are poor, some who are far from their place of origin, uncertain how to proceed, and possibly not even sure of where to look, and others who have never needed documentation, and should never need it, rural people who are surrounded by people who were there when they were born.
Naturalized citizens, while it helps to have a birth certificate, do not absolutely need one in the presence of other documentation. Legally admitted refugees are naturally often unable to obtain such documentation, and can get a green card without it. A valid green card is itself documentation for naturalization.
My wife, for example, born in Cuba, did have her birth certificate, which was used when she got her green card after having been a paroled refugee, but it was accidentally left behind at the US Consulate in Toronto. She was naturalized without it. Later, when applying for a passport, she decided to see if it could be retrieved to speed the process, and amazingly, after over 20 years, a diligent Consulate employee found it in the archives and sent it. But she was a citizen, and a voter, for years without it, and if it had not been found she'd never get it from Cuba now.
Of course, it ought to be sufficient to point out that the motivator behind this stupid and cumbersome rule is notorious for not accepting a birth certificate anyway. A large part of our government is run by tyrannical, demented racist lunatics and their cultists, and any expectation of how their proposals are likely to work or whether they will benefit the country ought to keep that in mind.
I just looked it up, and a Connecticut birth certificate, which is what I have, is $30.