Well first off tnis isn’t why they’re doing X, Y, or Z. Only what if this happened. But if you need a reason then perhaps this time they’re feeling a bit more paranoid or the Bush v1 administration was able to convince them through various means to join in. Or maybe Saddam was assinated and no one wanted an unstable Iraq falling apart with chemical weapons floating around. Pick your choice, but either way there’s a proper invasion and occupation.
Without a valid case for intervention, the coalition wil be an early version of 'the coalition of the willing' - you could look at Tony Blair to see how that how affected his reputation.
While I'm pretty sure that enough military strength could be mustered, the absence of permission for overflights, acceptance of the outcome, supply of fuel from gulf states, etc would hinder the build up, erode popular support, slow the victory. And if the invasion was successful, it's much harder to rebuild a society when neighbouring states oppose your ideas (including to the point of supporting insurgencies).
Personally I think it would have been better in the long run to finish the job properly the first time  but I see why they didn't.
 for that matter, doing a proper job the second time around would have been a good idea, rather than doing a bodge job. In my experience, things might not always be expensive for a reason, but they are always cheap for a reason, and Gulf War 2 was definitely done on the cheap.
So in short, simply saying that the original coalition wants to invade doesn't fly because an intent to invade changes the coalition, so another POD is needed to allow a pro-invasion coalition while meeting UN goals.