TGMR is in peril but that has been going on for a while. The ride is showing it's age big time and needs either major updates and refurb.. or.. more likely, it will be closed. But that's an unrelated issue to the name change.
Of the movies featured in TMGR - Here are their original studios of release.. (some of these rights may have changed hands)
Footlight Parade - Warner Bros Release
Singin in the Rain - MGM (at the time the ride was licensed)
Mary Poppins - Disney
Public Enemy - Warner Brothers
(Not Sure What John Wayne Movie is relevant)
(Ditto for Clint Eastwood - Dirty Harry?)
Alien - 20th Century Fox
Raiders of the Lost Ark - Paramount Pictures
Tarzan & His Mate - MGM (at least one of the iterations)
Casablanca - Warner Brothers
Fantasia - Disney
Wizard of Oz - MGM
So as you can see.. many of these movies come from studios other than MGM. (Note: No films from Universal! What a surprise

)
What is interesting is that I think as we near the 20 year anniversary of the opening of Disney-MGM Studios, the original licensing agreements all could very likely require re-negotiation. If that proved to be the case, I think THAT would be a bigger death-knell to the TMGR ride than the name change from MGM.
With all the changing of rights ownership over the years, and the original agreements possibly expiring or requiring re-negotiation.. that could spell trouble to the ride's long term prospects.
Just my opinion.
Knox