It could be cool if there was a franchise that had a new game by a different studio every 5-10 years. Like a "collaborative" franchise spanning multiple developers.
Bioware: Game 1
Obsidian: Game 2
Larian: Game 3
Owlcat: Game 4
etc...
I feel like they tried that with the new Star Wars movies and...uhhhh...those didn't turn out great.
I'd rather that a single studio stick with their franchise, expanding on their original designs/motivations/concepts in future games. And other studios create other franchises. This way, if the original game is good, the sequels are more likely to be of similar (or better) quality. And if the original game wasn't good, well then I probably wouldn't play the sequel by the other studio anyway...
(BG1&2 -> BG3 is not a good example of your suggestion, given the 20 year gap and the multiple changes of D&D editions)