You can't cancel
Turn (pivot) into
Squat (initial crouch) at all; you have to wait the entire 11 frames
Turn to finish before your crouch starts. You can't walk either or do anything else to get to crouch quicker from pivot. This is also the reason you can't pivot -> normal platform drop either.
So pivoting is only very marginally faster than wavedashing to get into crouch. To answer
Burglor
's question in the OP, there really isn't a fast way to crouch straight out of dash. The best ones are probably wd and waiting until run and then crouch. However just ASDI down is sometimes sufficient to get hitstun cancelled (which happens basically when you don't get knocked down and ASDI down).
Crouch knockback and hitlag reductions are effective in both Squat and SquatWait ("full crouch" state after the initial stage has passed) so the reduction starts immediately instead of after 6(?, I think it might be 7) frames has passed. Only SquatRV (Crouch release) doesn't have the reduction effect.