fiber mesh tape is good in my opinion, its more forgiving with wall movement than paper, so that should not be a problem.
I am leaning more toward the four step tape and bed process. Sounds like maybe the tape person applied too much mud at once and tried to do the three step process in one step. The mud was possibly so thick that when it dries and shrinks it has cracks like a dried up lake bed.
Before I go on, after the drywall was installed, how long was he/she there for the actual tape and bed work?