More specifically, basically the only absent 8-bit ops that have 32-bit equivalents in AVX2 are shifts and multiplies. Shifts are quite annoying (though, with a uniform shift they can be emulated on AVX-512 via GFNI abuse in 1 instr), multiplies are rather rare (though note that there is vpmaddubsw for an 8-bit→16-bit multiply-add). There's even a case of the opposite - saturating add/sub exist for 8-bit and 16-bit ints, but not wider.

GFNI is distinct from AVX-512; it was merely introduced in cores that also had AVX-512.