I had a similar experience. The hospital, I was born in closed about 30 years ago. The only birth certificate available from the state is a short form.Don't have an issue with states requiring documentation of proof of eligibility, but just a couple of comments...
1. Many states don't issue "long form" birth certificates any more.
2. A location of birth inside the United States establishes citizenship, requiring the hospital is not needed.
3. A location of birth inside the United States establishes citizenship, requiring an attending physician is not needed. (I hate to break it to many but many children are born with no attending attending physician.)
4. Just to be picky, "Natural Born Citizen" is a condition at birth and does not preclude gaining citizenship after birth in another country. Now attesting that there was no dual-citizenship status at birth is part of it.
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Also, some states, my own included, have destroy-by policies. Apparently there is no record of MY OWN long form medical birth certificate. I only know that because I've been carrying around a very old copy of my short-form birth certificate so long that it's seriously dog-eared, and last summer on a trip to Newport VT where I was born, stopped at the hospital to get a new copy. On an off-chance, I asked if I could have the actual record which showed my birth weight, ink blots of my feet, etc., and was told that those are automatically destroyed after 50 years because they serve no purpose. Operative words: SERVE NO PURPOSE. Hospital and state certification are all that is required.
Sadly, you will never be President
I guess I won't be able to make my run for the white house in Arizona.