You don't start a sequel by shitting on the original.
Not sure about that one. DS9 got its hero to a great start by basically telling the audience that the only apology he would accept from the original show's leading character would be a slow and painful suicide...
Had Kirk been the one to take the rejuvenating poison, I'd expect him to die at the end of the episode. There'd be no future for the character in a rejuvenated form, and no dramatic value in his return to his aged form. But the confrontation between Kirk and whoever got to represent the folks at Tyree's planet would no doubt have involved Kirk getting the final word and making his opponent repent, to the point of him delivering a touching eulogy. Picard would be the one left speechless in the final confrontation.
It's difficult to see how Kirk could have become more of a good guy in the story, though. And the very point of doing it would be the shock value of him not being one.