I believe the strategic thinking among the most influential foreign policy elites is that Gaddafi is doomed. He may win military battles but his victories will only force the rebels into a guerrilla insurgency which, under international sanctions, he cannot win. Does this sound like a good outcome? Not to these strategists. It would mean the rebels will have succeeded without Western military support. A no-fly zone will, naturally, be accompanied by "coordination" with the rebels including intelligence sharing and strategic advice, In due course, there will follow weapons and other military supplies and training. By the time the civil war is over, the rebel armed forces will be militarily dependent upon the West and our Generals will have become chums with their Generals. This is the intended result.
Besides, NATO is an enterprise supporting many military careers within itself and peripheral suppliers and organizations. It is in need of new missions to justify its existence. To a large extent, that is what the continued presence in Afghanistan is about, but that effort isn't going so well. Libya would be a second chance.