(Spoilers Main) What if Jon went with Tyrion?

When Tyrion was meeting with Lord Commander Mormont, he asked if Jon Snow could accompany him to Winterfell on his way back so that he could see his family and Mormont said no because the boy would have to forget his past ties.

Here's the thing: At that point, Jon hadn't taken his vows and was free to leave the Wall if he pleased. Mormont had no authority over him at all whatsoever as he wasn't a sworn brother as yet. If anything, it would've made more sense if Tyrion had approached Jon directly and asked him the question.

So, here's my question, what if Jon had left the wall and rode south with Tyrion while traveling down the King's Road? How would this affect the story?

Well, this is just a speculation of mine, but in canon, when Tyrion arrived at Winterfell, he received a rather cold greeting and immediately left after spending one night there. However, if Jon were with him, the welcome would've been far more accepting because Robb would definitely have been excited to see his brother. I don't doubt for a second that Robb would pull Jon to the side and tell him about everything that happened and how they suspected the Lannister's involvement. But Jon would vouch for Tyrion saying that the dwarf was with him the entire time and wasn't the type to kill children. And since Robb trusts Jon the most, he would feel more welcoming to Tyrion. So, Tyrion might actually spend more than one night at Winterfell before he and Jon leave. Hell, he might even miss Cat on the way, which of course would change everything. But that's just me.

What about y'all?