Looking around I find the primality of $n!-1$ has been checked using proth.exe above the range including $n=2022$. But I'm guessing this sheds little light on good was to search for factors of numbers of that form.

@hardmath Hi, do you also have a source how far the prime factors have been checked ?

@Peter I do not. PrimeGrid has an ongoing search for factorial primes, but it's not clear how to phrase the issue of factoring numbers of the form $n!-1$ that are composite. I found a paper on estimating largest and smallest prime factors of numbers of the form $n!+1$, but that doesn't seem to be germane.

Cf. this paper

@hardmath Seems small prime factors are known in factordb upto at least 2500 , I didn't find quickly new factors. Perhaps, we should run , say , 100 50K-curves on all the composites without a known prime factor.

I started running Alpertron on $2022! - 1$ using this laptop (the one I'm writing with at the moment). I didn't do a primality check because I was sure you'd done it (and $n=2022$ doesn't appear among tables of factorial primes).