Oh, I do think that Jon learning that he's a Targaryen changes his perceptions about his duties and, hell, the way he sees himself. He could consider the Throne. Especially if Sansa pushes him, considering he's still "family". But Daenerys can also see him as a threat to *her* legitimacy. These are basic reasons: the path to an actual betrayal would be more complicated than that.
He swore an oath to Queen Daenerys, and very publicly. That's probably the "duty" and honor that's primarily important to Jon. Feeling a duty to take the throne is just not in the cards for him, I believe. Are you arguing that Jon cares about royal bloodlines? I don't think we've ever seen him express that ideology in the show. Has he ever even mentioned that Daenerys is thought to be the rightful heir to the throne? That's been one of her key selling points, but as far as I remember, he hasn't used it to convince people that she deserves the throne. Instead, he says (in this ep) that she'll be a good queen. That's what matters.
Likewise, I have zero expectation that the discovery of his heritage will inspire him that his greatest duty is to mutiny Daenerys and take the throne because... they're both Targaryens but he's a guy so the throne has to go to him? That's... that's just not happening.
I can certainly see Sansa finding out and then trying to persuade Jon to take more power. The advisors might suggest this marriage idea. Sansa could get involved. Daenerys has no reason to cede power, so there's room for conflict. Jon probably just wants to push aside all this nonsense and fight the dead.
I'll go back to my other point: Sansa is not an idiot. And the show very much wants us to know that. She's not going to encourage Jon to mutiny before they've even fought the army of the dead. And I'll predict that Daenerys will prove herself to Sansa very soon, hopefully taking the air out of any such schemes.
So what does this Targaryen revelation mean to Jon, actually? I imagine it must be empowering. Legitimizing. He was never a bastard. He is the descendant of incredibly powerful people, and I'm sure that does mean something to him, in a self-actualizing sense.
Season 7 shines because it was so true to its characters. And I feel that in this premiere, too. Which is to say, the show knows deeply who its characters are and what they want. Daenerys wants the throne. That's always been her thing. Jon wants to defeat the dead. That is his thing, so much so that he only fought for Winterfell (his home and potential seat of power) because it might get him to that goal. With five episodes left, these two central characters are not going to get their wires crossed and change their goals. Which is also to say, they know what each other want. Jon knows that Dany wants the throne, to her absolute core. He would be insane — on a practical level, and on an emotional level — to challenge her on that. Not to mention that it would be at the expense of everything he actually wants.
The more interesting question is how Daenerys processes the information. I don't really see him ruling at her side, a never-ending threat to her bloodline legitimacy. But I'm not entirely sure how important the bloodline is to her at this point. Clearly you can simply take the throne if you have enough power. That's what Cersei did, without a claim, and it's also sort of what Joffrey did (with Cersei's involvement as well). The bloodline has some importance among the common folk, but I'd say it makes enough sense to them that a Targaryen is coming back, not the exact right Targaryen who has a slightly better claim (putting gender aside, Dany is a more direct descendant). Which only even matters if the truth of Jon's parentage is publicly revealed.
Clearly the best solution is for Jon to die heroically and not even deal with any of the politics. That would be a fitting way for him to go.
How many times does Jon have to tell you that he doesn't want the crown? I don't understand the premise of your point here — do you expect Jon to pursue the throne? He doesn't want any of this nonsense.
No, but I think Dany has become so power mad that when she finds out the truth about Jon, she'll preemptively strike for fear of him making the claim - even though we know he likely never would.
I think the revelation would rattle her for sure. It would be worrying, and a constant worry as long as he's around, no matter how many times he promises her he doesn't want to take her power.
I don't think she would straight-up assassinate him, though. That's nuts. I do NOT think Daenerys "has become so power mad." In fact, I'd say the text of what we see points us in the opposite direction — Dany has become more cautious and measured. For example, she barely needed any convincing from Tyrion to take the more humane route, to not melt King's Landing, and to not be "the queen of the ashes." Remember her conversation with Varys last season, when she asked him to openly and honestly advise her to keep her in check? Daenerys is anything but "power mad." She gets it.
Read about some of the absolutely brutal stuff Tywin Lannister used to do. Daenerys is a teddy bear in comparison. But people in Westeros viewed Tywin as a leader whose ruthlessness was practical. This is also a universe where the Boltons were allowed to hold power and ally with legitimate houses.
