<rant>I find this super-annoying. I'm not interested in spending time to promote Hyatt and that's one of the primary goals behind those mails.</rant> Do you guys feel the same or do you think you can somewhat move the needle in providing feedback?
Yes, I do feel they 'move the needle', at some properties more than others no doubt.
I don't disagree with one perspective they are looking for you to promote them, the ones I've gotten you can always just do the relatively easy survey part and skip the leave a review part, and hotel management still sees your survey. Finally, I think one aspect you're overlooking is that if there is an employee you would like to get recognition for making your stay better/good/great/bad, this is a great way to do that, it matters more at some hotels than others of course.