The claims to Gascony certainly were longer in tenure, so I take your point, but establishing a base of power there may be difficult, unlike with Normandy having proximity to Calais. Normandy also saw far more land claims of English families, while Gascony was a royal duchy.
As for England's nobility, its decimation didn't particularly make it all that capable of opposing a war that ostensibly it would benefit from. Edward IV was a powerful monarch, and quite frankly, Henry VII was less of a Lancastrian raising traditional supporters to win and more of a foreign conqueror using a mercenary army and being lucky to have the Stanley connection through his mother. The Lancastrians were decimated after 1471 and were a non factor at this point in time.
The war likely would have been popular because of all of the excess military manpower in England through the livery/retinue system that needed something to do in this period. If there was no war, and you have a weak child king with a relatively young, even if capable, protector, you will get brigandage to an unacceptable degree, very quickly.