Personally I think there is no benefit to you in giving "plenty of honest feedback". If you're open to staying if they make you an improved offer then you can try to politely explain your reasons in a way that leaves the door open for them to make such offer if they wish. But if they wanted to keep you your boss would probably have come to chat with you instead of calling an "exit interview" with HR...
Good luck on the new job.
As someone who is now in kind of a dysfunctional organization, I'm very thankful for people who give honest feedback to management. If everybody who leaves just says "It's not you, it's me!", how can management ever make informed decisions about changing processes and decision-making? OTOH I understand people's and my own urge to just play it nice, so people who will soon become just nothing more than entries in a list of contacts do not despise you. Especially if the scene of the city you live in is kinda small.
There's probably a middle ground here: give constructive feedback withouth dragging others through the mud. But it's hard to hit this middle ground in an exit interview that is scheduled on short notice. So what I'm doing right now, is prepping myself.