I think there were two basic issues here:

First off, it had never been done before in England. This was effectively the first colony outside of the British Isles peopled almost entirely with Englishmen that had "grown up" to an extent it could possibly consider running its own affairs. There was no real precedent for this situation.

When this situation came up later, in places like Canada and Australia, the British government knew what could happen, and had incentive to work something out.

Second, it was an issue of power and face. Parliament felt like they had the right to legislate for the entire Empire, and any kind of accommodation would necessarily entail politicians there willingly giving up some of that power.

Once they got rebuked by the colonists in the use of their taxing power, it was felt by the ruling party (the more authoritarian Tories), that they had to establish this power for themselves out of principle. As they kept trying this, the American colonists felt more and more like they had to resist the unrepresented taxation, again as a matter of principle. This instituted a spiraling cycle where both sides continually got more radical. Without compromise there was only one possible result, and every cycle brought the two sides further apart.^*

What the colonies were initially agitating for was something like the modern Commonwealth system, where the monarch would still be head of state, but the legislation duties would be carried out by a local elected body, not Parliament in London.

^{* - The details of this political death-spiral are described at length in Barbara Tuchman's The March of Folly.}