They probably didn't like the attention being drawn.
And speaking of our "willing coalition", most people think that the damage down to international relations involve only those naybobs that refused to help. Actually, I think it's been as much, if not more, damaging for our long term relations with our partners.
International relations is very much about political capital. I don't see that any of our partners were on the forefront of the push to overthrow Saddam. Sure the British went along, and they probably presented a better case than we did to the rest of the world. But I think it all goes back to the Americans siding with Thatcher in the Falklands. The British are willing to help us, even when we act rashly.
The support of America will likely cause a political shift in many of the worlds capitals. Germany shifted before the war - with the German head winning election exactly because he played against American intervention. Spain's government was changed - though terrorism and deceipt were the immediate reason. The Phillipines didn't seem too committed. Probably others will also teeter. Whether these governments joined the coalition or not, a majority of their citizens show to be against the intervention - including our staunchest allies - the UK.
And for those Eastern block countries, it's mostly an effort to show that they belong to the West - not that they care for the intervention. Which brings up another point - political capital. In order to build the coalition of the willing, the U.S. has had to expend a lot of it. Think those countries don't expect something for their support? After all, they are gambling that the payback for their support merits ignoring their short term internal populace. Someone's gonna have to grease those palms in the long term.
The conclusion I draw is that it's been a costly war in terms of foreign relations, even if you ignore the vocal non-participants.