But the deeper question is: why did they throw their eggs in that basket?
And that question is inseparable from the question in your OP.
Phil it's really not rocket science is it. Gaddafi was always the devils choice, even as Blair kissed his ass. The simple answer is that they gambled on the rebels winning and then the rebels went and spoilt it all by getting their asses kicked.
I also think there is an ideological dimension that has its own dynamics. Ever since Egypt both the US and European nations have been struggling to reformulate a new foreign policy that both accommodates the new emerging order while ensuring everything remains the same. Egypt is a classic example of this where elements of the old regime, the military, embraces the language of the democracy movement whilst ensuring their interests and most importantly from the West's point of view, their geo-political interests (yes Phil Israel) remain the same.
This has taken the form of a narrative and a set of assumptions One just as self absorbed as ever but one that presents the rolling wave of rebellions as an example of the primacy of liberal democracy. As such all national events are treated according to this narrative regardless of their particular national characteristics. They are all seen as desirous of recreating Western style liberal democratic regimes in the mould of the West. (again regardless of the reality of each case) The narrative dictates that the West must gently embrace these "democratic movements" even if it includes direct intervention as in the case of Libya.
The problem with this narrative however is obvious. It comes up against the reality of the fact that western power in region is based on the maintenance of thoroughly undemocratic regimes, particularly the Gulf states and those very gulf states are busy crushing democratic movement of their own most notably in Bahrain. This is a huge contradiction and judging by the number of comments across the world about Western hypocrisy, not one that is lost on people.
Libya is a dream for this narrative. It is a regime that the Gulf states hate. A regime that Western countries have long wished gone and a regime that really does itself no favours in terms of the Wests ability to spin it into the monster bogeyman of interventionist discourse. Gaddafi is the perfect candidate for James bondian villain. Really, you couldn't make it up. So, oblivious of the contradictions and hypocrisy, they march in regardless. Libya is the perfect world stage to demonstrate in BIG BOLD LETTERS the Wests democratic credentials. Supporting the rebellion but gently, no more Iraqs, no boots on the ground (yet) no direct regime change. Just " humanitarian intervention" to save the civilians. All aided by the genuinely heartfelt cries of "what about Benghazi"