I feel like I've heard this a lot, but it seems an exceptionally frivolous reason for such a monumental schism.

The papacy did not usually shy away from granting an annulment on precisely the grounds Henry asked for. The decision not to do so had a rather large political component.

I have to disagree with your basic premise. I can see your argument, although I don't agree with it, that the inability to dissolve a marriage was not earth-shattering for a private citizen.

However, in a monarchical political structure, where stability of power required male heirs, questions of fertility in the royal family were matters of national security, all the more so because of the bloody power struggle (ie. The War of the Roses) which had preceded the Tudors. The concerns over the fact that the pope would play a political game to the detriment of the integrity of the English crown is the exact opposite of a "frivolous reason".

Yes. Henry VIII was originally vigorously against the reformation. In 1521 he published a condemnation of Martin Luther titled Assertio Septem Sacramentorum. For it, the Pope gave Henry the title Defender of the Faith.

He made appeals to the Pope to annul the marriage. His reason was that Catherine was first married to his brother, but he died shortly thereafter and before the marriage was consummated (according to Catherine). She bore Henry several children, but all except one (Mary) died during birth or shortly after. Henry claimed it was God's punishment for marrying his brother's wife and cited Leviticus 18:16 and 20:21: "Thou shalt not uncover the nakedness of thhy brother's wife" and "If a man shall take his brothers wife . . . they will be childless." Henry claimed the Pope can't contradict such a biblical command.

On 18 June 1529, a Papal court was set up under Lord Chancellor Cardinal Wolsey to try the case of divorce. Catherine claimed only the Pope could do such a thing, but the Pope had been captured by the Emperor of the Holy Roman Empire and couldn't rule. Rome summoned Henry to go to the Rota, the supreme court of the Church, to try the case for real. Henry's court eventually ruled against Henry, however. He later fired Cardinal Wolsey as Lord Chancellor and appointed Thomas More (who took an indifferent stance on the divorce issue).

Henry's mistress and later wife Anne Boleyn was a secret reformer and influenced Henry toward reformation. She gave him William Tyndale's The Obedience of a Christian Man which claimed the Church nullified God's promises and usurped the magistracy of the prince. Henry read it and did a total 180 from his earlier position. He claimed Martin Luther was right to attack the vices and corruption of the clergy. Henry said the reason he originally attacked Luther was over just his challenge to the sacraments. Back in 1515 Henry claimed, "The kings of England in time past have never had any superior but God alone." In 1530, he realized he couldn't get a divorce through the Church and declared himself to be absolute Emperor and Pope in his own kingdom.

Henry was still skiddish about a total break from the Church and granting himself a divorce for fear of rebellion. Henry and Anne secretly married in 1533. He slowly chipped away at the Church's power which culminated in the Parliament's Act of Supremacy in November 1534 which separated the English Church from Catholic Christendom and announced Henry as the Supreme Head of the church.

Source: New Worlds, Lost World: The Rule of the Tudors, 1485--1603 by Susan Brigden.