Here's why Gettleman is probably staying.
The Giants have two of the top 10 rookies in the NFL this year according to PFF (Dexter Lawrence at 5 and Darius Slayton at 10). He also drafted Daniel Jones when the entire world mocked him for it.
He also cleared out cap space through trades where the Giants as of now have the 9th most to spend according to Over The Cap with another 15 mil available if/when they cut Ogletree, Ellison and Bethea...which would put them around 80 million available.
He also has already turned around a franchise like Carolina.
He has shortcomings no doubt (he doesn't trade down, which is infuriating, he made the Leonard Williams trade and lost a 3rd rounder because of it - although he may recoup a 3rd rd comp pick for Landon Collins leaving - and his FA signings thus far have not worked).
If the Giants want Rhule and Rhule insists he have some say on stuff, I'd think the Giants will be fine with it. Gettleman did work with Bettcher to bring in some former Arizona players to work with the defense he wanted, so it's not like he's a GM who doesn't look for input from the staff.
I would have been fine either with or without Gettleman getting the axe. But he's not as bad as some people are making it out to be.
If he brings in the right coaching staff (Rhule would be a great choice IMO) and targets all the money and draft capital on O-line and pass rushers, I think this franchise will be poised to make a strong move toward the top of the division in 2 years.