They might try. However, the USA has extensive counter-measures to prevent that. These types of missiles are not aimed but rather have some type of target acquisition system such as infrared (seeks out hot jet engine exhaust) or radar (follow signal bounce-back). Radar heads are probably too heavy for shoulder-firing so mostly infrared. But the USAF uses very hot flare dispensers on its aircraft which will cause the missile to lock-on to the flare rather than the aircraft.
Also, and especially after the Mogadishu “Black Hawk Down” incident in 1991, we’ve come up with offensive systems to lock onto and eliminate personnel with these types of missiles. Usually they “paint” the sky with a sensor and lock onto an aircraft before firing. So we lock for their “paint” system, and then an AC-130 Gunship, miles away, can direct precise 50 caliber fire to that “painter’s” location. Wax one Jihadi.
Even without all that, I’ll bet you aour guys were foaming at the mouth wanting to go. Too bad our leadership was such pussies.
Two other points just to clarify.
First, the Mogadishu shoot downs were using RPG’s, which are indeed aimed and fired. Still, the issue that was fully researched was to deal with all shoulder-fired weaponry and improve our survival through mediation of these weapons. There are other methods for RPGs...
Second, Kadaffi’s weapons were all old Soviet or European (mostly French) stuff. NOT as good as US Systems like the Stinger. We’ve had years of testing and experience with the Soviet systems and even French systems used by Saddam’s forces in Iraq first Gulf War.