Ser Davos has no ties to The North, only to Jon
It is not explained exactly what it is that ties Ser Davos to Jon Snow, but he definitely seems to admire him, as a leader and also as a person.
When Jon brought him along to Winterfell, won the Battle of the Bastards and was named King in The North, he didn't actually proclaim Davos as his Hand, or any other official position. So there's no reason for any of the Northern lords to follow Davos.
There must always be a Stark in Winterfell
What, like name a random southerner, up-jumped to his position by a 'traitor King' to rule over Winterfell and the North over a Stark? No way, there's no chance that the Northern Lords and Ladies would have that, no matter how much they support Jon.
The true war is coming
Additionally, when Ser Davos read the letter which was sent by Lord Commander Mormont asking for assistance against the impending doom of the White Walkers, he was instantly inspired, even convincing Stannis to take up that cause as well. This is Davos' primary objective, and going to Dragonstone to get obsidian and perhaps even the dragons themselves serves this purpose.
Even if he were the Hand
Remember when Jon announces that he must go to see Daenerys, and Sansa says:
Then send an emissary, don't go yourself
Daenerys is a Queen, only a King can convince her to help us
What he's saying is that he needs as much power and authority backing him as possible, it would stand to reason that he would take his supposed Hand. Besides, Dany has her Hand next to her, why not Jon?