Can I convert a tourist visa with a 60-day COVID extension into a Non-B visa in Thailand?

whether you can or can't go from a tourist visa entry to a 90 day in country Non-B is totally between you, your future employer and the immigration office you deal with.

YES, many people have done it. It comes down to whether your employer can meet the documentation requirements to get the 90 day Non-B issued in country at the immigration office. Their requirements are much more stringent than the requirements to get a 90 day Non-B from a thai consulate outside the country.

Talk to your employer, they're the people who will know