Hospitals aren't issuing documents themselves, they send notification of the birth to the local authorities who issue the documents. The problem is when the state tells the locality not to do that without checking the parents immigration status and the hospital notification means nothing.
The local authority still has to issue a document whether the status is checked or not. Some 3rd country isn't going to grant citizenship even if eligible if there's no documentation of birth.
How is, hypothetically, a locality in Texas going to issue a certificate if Texas passes a law not to? If they issue some "non-documented parent" certificate, Trump's order will prevent issuing a Social Security number, etc. It's chaos.
It would be chaos until things corrected at some point in the future, as long as there's a paper trail.
Kids born abroad are US citizens if 1 or both parents are US citizens. They will need a birth certificate from the foreign country to apply for US passport. It's up to laws of the country of birth to determine whether they're citizen of that country too.
Trump's order purports to have the Federal government administratively ignore laws like that. Citizenship means very little if the government doesn't believe in it.
If you're a billionaire living permanently abroad, the IRS would be very interested if you have a US citizenship.