It might be true for the fact checking groups described here, but to state that as a general case is incorrect.
> Fact checking is better referred to as "authority enforcement" or "mainstream validation".
Fact checking at its core is about analysing assertions that can be objectively verified.
Fact checking must bring forth strong references/sources or proofs to either go in or against the initial assertions.
> Here's how you know the fact checkers are actually propagandists: you can't find their name or their credentials, or who they work for/paid them.
If a fact checker does not provide this kind of info, then yes, it does not seem very trustworthy. Although you could still evaluate them based on the sources they are using for their fact checking.
But, obviously, it does not have to be the case. Fact checkers can and will provide info about their funding, affiliations, composition, etc...
As an example, you can take a look at the FAQ of this fact checker that was active during the French presidential election of 2017 https://crosscheck.firstdraftnews.org/france-en/faq/ :
As you can see, they clearly state that they received funds from Google News to set things up. They also share details about their partners, members, etc.