A broad coalition of unequal partners...
makes tragic "friendly fire" incidents all the more likely. As it is, all too many Americans will be killed due to mixups in communications and human errors. Throw in incompatible equipment and language barriers and you'll probably have even more mistakes and deaths.
Besides, soldiers don't fight for flags or slogans, they fight for their buddies. A soldier wants to know that the man beside him will not let him down. You just can't say that about the French. They don't have a very good combat reputation, and them failing to hold up their end of the fight would get more Americans killed.