Birth Certificate is one thing. Medical records are not quite the same thing. They included lots of names/signatures and documented medical data/prescriptions, dates, times, et all. And her physicians have signed off on all of it. No way she could get away with fraudulent med records.
suuuuuuuuuure
Even if some of her medical records were faked or deleted, her biggest problem is her prescriptions. My guess is that she takes one or several prescriptions for various problems. (Most anyone her age is likely to be taking something. I'm her age and in excellent overall health, but even I take three prescriptions for a minor problem.)
It would be extremely difficult to hide prescriptions. Each one has a long paper trail, and a "cover-up" would be politically devastating if it was penetrated. (Trump likely knows from his private investigations what she's taking.) And each prescription comes with a long list of precautionary warnings. (For example, "Can cause headaches, nausea, elevated blood pressure,... Should not be taken in combination with these other medicines... If you have a heart condition, liver condition,... consult your doctor before taking.") We've all laughed at the laundry list of warnings in every drug commercial.
If Hillary reveals the prescriptions she's taking, everyone will immediately search through the list of side effects and wonder if she's suffering from them. If the prescriptions are sometimes used to treat brain conditions (such as from her concussion), everyone will further speculate on her mental and physical health. Whether or not any of that is true, it provides confirmation bias.
Unless Hillary really has a fairly clean medical history, she's in deep trouble. The ground has been laid over the past few weeks (and months) to question her health, to the extent that she had to open a (pre-opened) pickle jar on national TV to try to laugh it off. Now if she stonewalls her health records, nobody will believe her protestations that she's healthy. And if she releases her health records, she'll provide more fodder for her critics.
It's a brilliant move by Trump. Maybe not checkmate, but certainly a highly threatening gambit.