What was the "martyrdom" of Thomas Beckett really about?

Now, in popular culture, the martyrdom of Thomas Becket is presented as a fairly simple thing. Thomas Becket was a commoner from a prosperous background, who, after obtaining an excellent religious education, advanced through his ability to the position of archdeacon of the Archbishop of Canterbury. As a result, he was then recommended to the still very young King Henry II as Chancellor, and he served this role with distinction, if with some corruption and extravagance. Then, as they both got older, and the Archbishop of Canterbury died, it was perfectly natural for Henry II to have his friend and advisor elevated to the position of Archbishop of Canterbury himself, particularly to advance his own agendas of centralizing power under his control, something Thomas Beckett had thorougly supported as his chancellor. Then, something rather strange apparently happened. Thomas Becket suddenly "got religion", to an extraordinary extent to which led him into direct and fairly violent confrontation with his heretofore master, Henry II. Now, in popular culture, this conflict is generally represented as being little more than public argumentation, in which Beckett puts forward the case that God cannot be controlled by King, so the Church must remain independent of the State. And, again in popular culture, Henry II eventually gets tired of this public arguing, and makes an indiscreet remark to his knights at arms, "Will no one rid me of this meddlesome priest," which leads to the public "martyrdom" of Thomas, very much against the will of Henry II himself. Actually, the truth of the matter is rather different. This conflict was highly political in nature, and, actually, perhaps a bit counter-intuitively, the Pope himself was by no means a supporter of Thomas Beckett in the position he adopted. Indeed, if the Pope had supported Thomas, Thomas certainly would not have been martyred, and Henry would almost certainly have been forced rather quickly to submit to Church independence, something he had to do in any case, following the martyrdom of Thomas. Now, in many ways, the conflict between the Pope and Holy Roman Emperor Fredrick Barbarossa mirrored the conflict between Thomas and Henry. However, while Fredrick was in an excellent position to invade Italy and threaten the Pope, Henry, with his power base primarily in England, and with the French King between his Western French dependencies, was not. Also, Henry's Angevin Empire and the Holy Roman Empire were competitors for power in Europe. So, actually, Henry II of England, was a supporter of the Pope in his battle for Church power, with respect to Fredrick, so, actually, the Pope was disinclined to support Thomas Beckett's aspiration's to essentially the same power, in England! Hence, to describe this conflict as being effectively about God, or the Church, is a gross oversimplification. It was more a conflict about pure power, itself, or, more specifically, about the balance of power, in England specifically. Thomas Becket virtually set up his own mini-Kingdom in Angevin controlled France for a time, after having escaped from custody after being convicted of embezzlement, and was a fleeing fugitive when he arrived in Norman controlled France. For years he engaged ina cat and mouse power game with Henry, excommunicating English bishops who opposed him, and was thought by many to be trying to actually make himself King. Eventually, he did return to England, after a half-hearted "reconciliation" with Henry. Inevitably, this just escalated the level of confrontation, until Henry finally ordered him arrested. When he refused, along with his armed guard, they killed him. Nevetheless, this "martyrdom" discredited Henry with his own people, and, did result in continued church independence in England for three centuries, until the Reformation of Henry VIII.