Looking at the Empire Strikes Back crawl, it says that Luke Skywalker led a group of freedom fighters to Hoth - which when you consider A New Hope… surely that role would’ve been assigned to Princess Leia?

Next the crawl it says Vader was obsessed with finding Luke Skywalker. Why? I would’ve thought he would be seeking revenge against the Rebels, as a whole, for blowing up the Death Star? During the time Luke fired his torpedoes, Darth Vader only knew that “the force is strong with this one.”

It seems to me Vader is not given permission to find Luke Skywalker until when Vader chats to the Emperor. He knows he has a son but it’s the Emperor provides him his name. Though this contradicted when Vader had earlier in the movie said "That is the system! And I’m sure the Skywalker is with them. Set your course for the Hoth system. General Veers, prepare your men.”

So does it make more sense "That is the system! And I’m sure the Princess is with them. Set your course for the Hoth system. General Veers, prepare your men.”? Vader had met the Princes in the previous movie and he knew she had, at least, a significant role in the Death Star’s destruction and had a vendetta against her?

Am I missing something?