I'm 100% sure that getting EUA certification has everything to do with having the proper documentation in place and proving to FDA that you do. FDA is a good organization, and I know they're doing their best under very difficult circumstances right now. If they start letting everyone do Covid tests without documentation, they're opening everyone up to lawsuits when John Doe tests negative and he goes out and infects Jane Smith's grandma who then dies. I'm also 100% sure getting EUA cert doesn't mean that your test is necessarily that good, but rather that you have a good handle on its sensitivity and specificity. For what it's worth, if your test isn't CLIA certified, you can't get reimbursed for it, so it's a big deal to hospitals too.
I am so so grateful for everything the FDA is doing these days. It just makes my life harder that I can't accept any test that comes across my desk. But I know those poor bastards are working an insane amount to verify as many tests as they possibly can to alleviate the stresses on the current approved tests.