It’s more likely that they are being sent because they are trained to operate THAAD and sending them is more efficient than training IDF troops to operate it.
Iran launched missiles at US bases in Iraq a few years ago with very little in the way of retaliation.
Yeah I think that’s why Iran attacked the US base (might have been multiple, I can’t really remember) iirc.
I think a big part of why the US didn’t respond is that a) they took out Soleimani first and b) an Iranian air defence battery, on high alert expecting US retaliation, immediatley shot down a passenger jet.
The airliner being downed really took the sting out of it, kind of like someone swinging at you, barely connecting, and then immediately punching themselves in the dick.