This would no more violate the ADA than random drug testing (legal) or requiring temperature checks, which some employers are already doing to protect against COVID-19.
Also, the ADA is not a constitutional amendment, so even if there is a potential conflict with ADA, Congress could easily address that in a new bill without needing to pass a new constitutional amendment.
The Supreme Court has repeatedly held that the government has a pretty wide reaching authority when it comes to fighting disease. If the federal government can quarantine entire towns and force vaccines, they can similarly force you to be tested.
They can't quarantine towns from the federal level. Not in the least. A state might be able to, because the 10th amendment is a thing, but Federal government cannot. And neither state nor federal government can force vaccines because it violates the 4th amendment. You don't get to search my body for diseases and use them to discriminate against me. Sorry. Furthermore, Roe v. Wade was argued on the basis of a right to privacy that's implicit. So no, I'm right on this.
I mean, you’re entitled to your opinion, but at least some people disagree.
Edit: and, your claim that there is no such precedent is a flat out lie.
I am so embarrassed on your behalf.