If memory serves, I don't think it was Wilson that was pushing for a new contract. He still had two years left.
I believe what happened was that the Walton's bought the team and decided to come into the NFL like a bunch of big swinging dicks by putting their (mushroom) stamp on the franchise ("oh, look at us, we're rich and powerful and making an impact").
I'll try and find a source for this. As much as George Paton should get roasted for the trade (and Nathaniel Hackett), I believe he was at the mercy of the new ownership group when it came to the contract.
I also don't think more teams would have been in on him. There were people saying he was in decline even before he was traded by the Seahawks.
Edit for links:
Report: GM George Paton Safe, Wanted to Wait on Wilson Extension
Inside the Russell Wilson-Seattle Seahawks drama that led to the Denver Broncos trade
Russell Wilson’s prime is over