But that is a distinction without difference. In practice, if you cannot access your birth certificate, if questioned, you cannot prove you are an American citizen.
Correct. But that also goes to my point above. I am in that situation (I was born in Canada while my father was stationed up there in the Air Force). I recently relocated to Florida while my foot is healing, and got a Florida Driver Licence and registered to vote. I had to show all sorts of certifications, including my birth cerfticate, the certificate that proved I was granted US Citizenship because my father was in the service and out of the country at my birth, and letters from the US embassy at the time. (Yeah, even for Canada). All of that however, is based upon that original birth certificate, without which, none of it is accepted.
So if a birth certificate is withheld, the state is definitely refusing to grant the baby US citizenship. Technically, they can say they aren’t, but without providing the documentation, they are. Without a copy of that, they will be refused a Passport, a Social Security card, eventually a drivers license and of course a voter registration card.
The problem with matriculas, however, is that regardless of how shoddy a job that Mexico does in verifying them, they ARE official Mexican identification for Mexicans living outside of Mexico. They are used for countless legal documents as verification of identity, including many legal documents involving interactions with citizens of this country. Marriages, trade contracts, etc. If Texas is allowed to reject IDs for this specific reason, it seems to me they have to reject them for all reasons, which invalidates each and every document that relies upon them. And that’s going to be much, much bigger headache for Texas.
Just last September, Congress voted down a measure preventing their use by US financial institutions. So, the US Congress has accepted that they are acceptable for conducting business. Many (if not most?) other countries allow their use, Peru for example, specifically cited their acceptance by the US Government as the reason they accept them.
And they have been in use for something like 150+ years. So it does seem raising a question about why they are a concern now, for this specific usage, is now a problem.