Frankly I would leave a role well before I'd send unsolicited, negative feedback over my manager's head. Even if my feedback was well evidenced and actionable, and my director took action on it, I don't think it'd do me any personal favors.

I'm not a manager, but I expect you get there through strong relationships and trust.

Going over somebody's head indicates you don't have a strong relationship with that person, and that you can't be trusted by a direct superior.