"Rome constantly sought to override the Sovereign in state matters."
Quite right and true whether or not one sees Beckett as a saint.
For details on all of this simply google up the Constitutions of Clarendon that Beckett and Henry II fought about. It really had nothing to do with Beckett fighting for the people. It had to do with ecclesiatical privileges v monarchical privileges.
In an Erastian system such as launched by King Henry VIII, there could obviously be no more pilgrimages to the shrine of a person who had challenged royal authority. Remember that Herny VIII was only a "Protestant" in a very narrow sense. He defied the Pope of Rome. But as to theology, he was hardly on the same page as his Archbishop of Canterbury, Thomas Cranmer. But inasmuch as he had defied the Pope of Rome, the shrine with Beckett bones had to go.
Now, that is not a bomb, it is merely compost.