I am afraid your history is a bit off.
The British did an excellent job of working with bad (or good, depending on your point of view) Muslims. And they did it very cheaply, for the most part. Their last hurrah was their defense of Oman with special forces and mercenaries.
The Muslim problem is one that requires active management. Consider it pest control. And the people who deal with them certainly can’t be fools. There are special talents required. And they need to be on the spot.
As for what happened to the British, and all the other colonial powers, and for that matter, even God - WWI.
Before WWII the muslim brotherhood, circa 1920’s, there was no significant active caliphate restoration planning.
There was no danger of muslims moving into Britain in the 1800’s and subverting the entire nation to submit to islam.
Christianity was still too prevalent; as an example, note the popularity of Charles Spurgeon during that time.
There also was no technology so advanced during the 1800’s that whole cities could be destroyed by a small band of marauders. There was obviously no jet airline travel, so dealing with islamic countries was akin to dealing with barberic natives that had no hope of attacking the home country.
The British militarily in their colonies during the 1800’s simply “carried on”, this management style you speak of was simple and effective during that time in those places, as there was no massive homeland threat as there is today to Western nations.
I agree that the British control of colonies was quite effective in it’s day. IMHO, India benefits from this today in at least one way, as it’s citizens, paired with an excellent educational system, produce great results.
In the post WWI negotiations, the European nations and America made the colossal mistake of even giving muslim leaders a seat at the table. Most of the ruling intelligentsia types who were the brainiacs behind the negotiations for the victorious powers reasoned that rolling off their colonies into new “nations” would allow the transacting of commerce with little puppet leaders put in place. The empires, one can only surmise, thought that they could have the benefit of commerce without the cost, complexity, aggravation and general unsavoriness of having this colonial domination.
The post WWI negotiations, in their birthing of new nations, were way off in their thinking that “people are basically the same all over the world”, and the vastly underestimated the complexity of international relations that would be realized with such enormous political changes happening so quickly. Not to mention that German agents had also used the same tactic they did in WWII, of generating anti-British sentiment in the middle east as a way of diverting British war resources to the colonies. In the post WWI era of the 1920’s the muslim brotherhood sprang up, and anti-Western sentiment has been growing ever since.