It's not a question of legitimate annulments.
It's a question of whether a sacramental marriage ever existed.
No annulment can void a valid sacramental marriage, no matter how much you want to dance around it.
Granting Holy Communion to unrepentant adulterers, even without the paper "annulment", may give the sinners great peace of mind and soul. But it's a false peace and it's a mortal sin.
I agree “if” it was a valid marriage. That is why you have the full process in certain cases.
If outside the Church, is just doing the paperwork needed. Also those who simply did not wait, there a practice called conflagration of the marriage.