I think the point of Jon making all the houses band together for him before anyone discovers his real identity is that they would probably go
![ehh :ehh: :ehh:](https://www.thecoli.com/styles/default/xenforo/smilies/ehh3.png)
at him being the rightful heir, regardless of last name.
Now, him not wanting the gig is an interesting point because even though he accepts responsibilities he doesn't want throughout the series, his purpose is making Westeros survive the Long Night. If he comes out of this alive, I'm not sure he would see ruling the Seven Kingdoms as a responsibility he has to accept for the "good of the realm".