The Percys were descendants of Edward III through Elizabeth Mortimer. When Henry VII took the crown, he and his son, Henry VIII, seemed quite ready to eliminate anyone with a possible claim to the throne (Pole, Stafford). Yet, I never hear of the Percys, despite being from a powerful noble family and with as good of a bloodline claim as the others, including the Tudors, being considered as possible claimants or heirs to the throne.
It is important to note that several clans in the peerage had a logically better claim to the throne than did York, such as Somerset or Beautfort.
Only Richard of York grew incensed enough by the mis-governance of Henry VI to risk his life and challenge the Lancastrian regime, and only his ties to the Neville family gave him sufficient clout, both in Northern and Southern England, to win. But York was hardly the clan with the most direct tie to Edward III.
As to why the Percy's survived the purges undertaken by Edward IV, Richard III, Henry VII & VIII, I have no real answer save that the Percys were both lucky and relatively loyal to their chosen faction: Lancaster until Warwick Neville joined it, and then York until Richard III alienated just about everyone. They had their share of executed members, particularly during the first phase of the War of the Roses. In fact, House Percy was simply lucky to be rehabilitated to their old powerbase in the North by Edward IV during his efforts to diminish Neville influence.