1.) Part of the reason I was thinking about this was based on a discussion I had about DM's with favorite scenarios to run. A lot of us have favorite scenarios that the more we run them, the more we are able to weed out the issues, make the NPCs a little more interesting, and make the experience more fun for everyone. So, part of the idea was that it would benefit everyone to help encourage DM's to master a few scenarios (moreso than how it currently works) by helping to incentivize them for doing so.
2.) Off all the benefits, I think getting extra access is probably the one I find most appealing as an extra reward. I'd run games without it, but still this is probably the one I like the most.
3.) I'd say, as I DM more often than I play, probably about 80% of the time I reserve this to save characters from death or a condition that will remove them from play for a while. It's nice for that, but I get my Reroll from things I've purchased. The +1-+3 is pretty irrelevant for the uncommon personal use. It's the reroll itself that's the good part, and that has nothing to do with GM Stars. Something that's also worth noting, simply having DM Stars does not give you a free reroll. You need to have one of the products to get the free reroll, and then if you have DM Stars, you get an additional bonus.
4.) I've had this from the start and never once felt any need to use it on any character. It's just, well it doesn't do much for the divine classes I mostly play.