I don't believe the FDA would approve just a paten

I don't believe the FDA would approve just a patent, they need to see the test, the packaging, where and how it will be consistently manufactured in an approved setting and at scale. As things progress if someone else whishes to manufacture the test that facility would then also need approval.

I think you can look at Therma Bright as an example, not comparing the tests, just as a small company trying to brake through. Dec 1st PR said this,,
"is pleased to announce it has signed an agreement with a contract manufacturer to produce a weekly minimum of 500,000 AcuVid™ COVID-19 Rapid Antigen Saliva Tests in order to meet the new U.S. Food and Drug Administration's Emergency Use Authorization (EUA)"
"We're excited to add a new manufacturing partner to our Therma Bright team; one who alone can help us meet the minimum 500,000 weekly production requirement per the November 15, 2021 FDA guidance for Covid-19 antigen diagnostic tests developers,"

I'm just an observer making up stuff. Don't go to the bank on what I say!