You don't know much about the ME. The UAE ruling parties would lose all influence with their people if they didn't defend the regions honor. You simply don't understand what the implications are. That is why so many active and retired military leader types are being outspoken about this.
We've been dealing with those governments for years. Saudi Arabia had no problem taking our troops in when Iraq invaded Kuwait did they? They needed us militariy protection and so does the UAE.
I'm not disputing any of what you're saying regarding internal posturing and possibly oncreased anti-Americanism, which they'll overcome with force if need be however at the end of the day the government will get over the rebuff simply because they'll have no choice.