Does no one else agree that the Doctor being immortal would undermine the show?
Depends on how you define "immortal." It would certainly undermine the show if there was no longer a possibility of the Doctor being fatally injured before he could regenerate in the course of an adventure.
But I don't think that it undermines the show if the Doctor were to be otherwise immortal. Really, the show has
already made him
de facto immortal simply by virtue of the fact that they don't really address the idea of a regeneration limit and by virtue of having already made him so much older than all of his human companions. If they were to slip in a "the destruction of the Time Lords lifted the artificial limit on regenerations, rendering the Doctor immortal barring injury" infodump, it wouldn't realistically change any of the show's thematic content. It's already fundamentally a show preoccupied with the idea of how an immortal man copes with living in a universe of comparative mayflies.