I think the "Peterman hate" boils down to a few camps.
First you have the "not my QB" camp. They are the people who wanted Watson, Mahomes and Kizer types. They take exception that he's not the guy they wanted and nothing is going to change that until Peterman shows he is a good QB.
Secondly you have the "Threat to Tyrod" fans. These are the ones who no matter who we drafted would be at that QB because they are threats to steal his job and kick him off the team, it's less pronounced now then before (largely because of his own undoing) but that was a very vocal anti Peterman camp especially early.
Thirdly you have the "instant gratification" crowd. The you are only as good as your last drive types. To them it doesn't matter that you are a project, rookie or game situation. If you aren't lighting it up then it's because you are awful.
Outside of those three groups you are going to have a few minor ones mixed in with the Boxscore scouts, Style, Tools and draft location people. These are the ones where no matter what he does (even if he becomes Brady 2.0) will always hate him no matter what because of what he's not or where he was drafted.