He gave his word to many people. His only loyalty is to the realm, as he said.I find Varys plotting against Dany a bit weak. He gave Dany his word that he would serve her after supported Robert in trying to assainate her, and now yet again he's plotting against someone he's sworn to serve.
Obviously he is desperate to say it — being King of the North doesn't make him a Stark, this does, and it's something that he had been craving for all his life. But I agree with your first point. The whole plot of Dany's first 5 seasons or so was about her becoming a queen instead of a good-hearted girl, making hard decisions etc. Jon hardly ever ruled, his whole "rule" was dedicated to one project — introducing wildlings to the world. He'd be more ruthless (and... bad in their eyes) if he had faced the same obstacles.I don't understand how Tyrion and Varys can be convinced that Jon would make such a great ruler of westeros. He has a good heart, but is dumb as feck. Dany was absolutely right, when she wanted Jon to keep his heritage a secret and not tell Sansa and Arya. He should absolutely logically be able to see, that it's the best course of action so far so why is he dying to tell his family. He's been a bastard all his life, then king in the north and later warden of the north if Dany wins. Or they could do the completely logical thing and just marry. I don't see why he is so desperate to tell anyone that he is targ.