I used to wonder about this as well; however, thinking about it, it seems to suggest itself. Since the objective here was to make the people of Ayodhya perfectly happy (since his rule was supposed to be ideal and perfect), his quitting the throne would have caused public unhappiness, and therefore have defeated the purpose of removing her from the throne in the first place. Otherwise, you are right - even Lakshman, who declined the yuvraj, would have been happy to swallow that poison for the sake of his bhaiya's and bhabi's happiness.
As for why he took the subjects with him, I agree with you on this, but have one theory why he did so. Ethically/morally, Kush and Luv wouldn't have been comfortable ruling the same people who had insulted his mother, because that would have been an insult to her memory in and of itself. Rather than put them in such a dilemma, Rama decided to solve that by taking them with him, so that none of his sons or nephews would have to accommodate them. So he did what he did, but I still think that allowing them to accompany him to Vaikuntha was a little much - after all, Vaikuntha belonged to the same Sita whose chastity had to be proven, and therefore they certainly didn't deserve to end up there. I do wish once the transfer was made, Vishnu had Yamaraj come and pick them up, and take them to where they belonged - Yamalok.