Maybe on some models - I've been doing it for years and I have no issue.

I don't put a ton in, and I imagine most of it is neutralized by leftover soap residue and a relatively hard water supply.

It will accelerate corrosion and rust. Modern, higher end machines are probably going to use stainless for all the affected parts but I wouldn't risk it on an expensive newer machine. Older machines are more likely to have non-stainless parts and if you've got an older machines you probably don't want to accelerate its demise.

Wouldn't chlorine bleach also have the same effect on your washing machine since it is quite the oxidizing agent?