I don’t think she’d want the power to like, shape the universe in her image~ or whatever. Actually, that gifset cuts out the bit that I think would be tempting to her. Immediately before the dialogue in the gifset comes in Giles addresses Rose and says:
And you could be with him throughout eternity. Young… fresh… never wither, never age… never die.
And then back to the Doctor with
Their lives are so fleeting. So many goodbyes. How lonely you must be, Doctor.
I think in wake of everything School Reunion drags up — the fact that the Doctor does always leave or otherwise lose his companions, the fact that even if Rose does spend the rest of her life with him he can’t spend the rest of his with her, and Sarah Jane standing there as living breathing proof of all of this — they’d both be tempted.
The Doctor mentions the Time War in this scene, and I’m sure that’s part of his motivation, but it’s worthwhile I think to note that it’s not until Giles brings up the prospect of Rose ~never withering~ that the Doctor looks particularly tempted. They could have rearranged the dialogue — had Giles bring up Rose first, and then Gallifrey — to put the emphasis on the Time War, but they don’t. And considering the rest of the episode, and how it focuses on companions leaving and what happens to a companion after they leave, I think it’s safe to say Rose is a huge part of what’s tempting the Doctor.
Plus it’s Sarah Jane who objects in this scene, both in the speech in the gifset and earlier (“Doctor, don’t listen to him”). Rose is quiet. It’s not that I think she’s power-hungry or anything. But just a couple hours earlier the most important relationship in her life just got tossed on its head, and she was confronted with all these roadblocks that I don’t think she was even aware of before. The idea of being able to make them disappear must be pretty enticing.
It’s not that I necessarily think that if Sarah Jane hadn’t been there, the Doctor and Rose ABSOLUTELY would’ve taken Giles up on his offer, or anything. It’s quite probable one of them would’ve snapped out of it before then. But I do think it’s significant that the reality check here comes from Sarah Jane, who’s already experienced this loss and discovered “some things are worth getting your heart broken for”.