At the point where I have to get Gothrens' vote, he keeps telling me over and over that

he will take it under advisemant or something like that and never gives me an answer.

I went to a game wiki and found out that you have to kill him so I go back and whack him

out and now my quest journal won't update past my waiting for an answer from him.

I can go to an earlier save and kill him on sight but I want to know if he can actually be

persuaded peacefully to vote me in?

Everything's fine, just talk to any of the councilors after killing Gothren and you'll be named a Hortator. If you want a special reward, talk to Aryon.

This. Also, no, I don't believe there's any way to get him to come around. If you want to be Telvanni Hortator, you have to kill Gothren. (He is a jerk, though so it's not terribly difficult to justify!)