Hmm. The choice wasn't really open to him. Only Yue really liked him for the job and Arnook had him thrown out. If Arnook had listened to Yue and made Zuko heir, Zuko at this point is still in his "Fire Nation or Nothing" mentality, and he would have been a sulky, resentful, horrible ruler. Depending when Arnook died, if Zuko had stayed and tried to open negotiations with thr Fire Nation, he would be opening negotiations with either Ozai or Azula, who are going for world domination. Zuko holding the NWT would be a personal affront. Resentful as Zuko would have been, I'm not sure he would have done anything about the gender equality of the North, just because he'd be too busy sulking. Also, it's unlikely he'd keep the throne for long. The NWT would rebel.

And Lu Ten and Rinzee would be stuck brainwashed in the Earth Kingdom.