It's a fair question for a civilian to ask, but in practice, hollywood doesn't reflect reality. This is not a Michael Bay film.
That said, do you know how many meetings were held in the past week where someone who should have known better asked the same question and because of his or her rank, the rest of the people in the meeting had to get on the phone and ask the people who just might, maybe, on the off chance someone knows something. SpaceX was probably called at least four times, someone probably demanded to talk to Elon.
There is at least one US proven aerial catchment system in existence, but it was not designed for this, and likely it has been mothballed for so long that by the time they even figured out where the dang thing was the balloon would be gone, much less figuring how to modify it to catch a balloon satellite.
https://en.wikipedia.org/wiki/Lockheed_C-130_Hercules?wprov=sfti1
The aircraft was the JC-130. There is a photo of it in the process of catching a satellite film canister (probably simulated) in-flight about halfway down in the C-130 article linked above. It was a temporary conversion according to the article, so resurrecting it from mothballs was a no go.
I seem to recall that the mystery ballon was at about 60k feet when first reported. Not many details reported on the shoot down (Altitude, weapon used, aircraft, etc.). The only details I’ve seen about the shootdown is that the fighter was out of Langley and that the debris field covered 7 square miles. (The latter detail being used by the Dems to justify not shooting it down over land.).
Other Freepers have posted about the difficulty of getting a lock on the ballon itself. That seems reasonable. File photos of inflated instrument ballons show them to be huge in size. Early reports said the instrument package on the Chinese ballon was about the size (in volume) of a school bus.
Since both are well chilled by their long time aloft, there probably wasn’t much for an IR sensor to lock on to. Might have had to use a laser guided missile and separately illuminate/designate the target. Even then, it might have been necessary to light up the payload (despite wanting to recover it) in order for the missile’s seeker to “see” the target.
Well, a lightweight instrument package hitting the water at terminal velocity from altitude wasn’t going to stay intact anyway. Probably helps explain the reported size of the debris field.
</ my two cents>
Update to my previous post:
A lot of detailed information about the aircraft, weapon, mission execution in the video. Also excellent information on the Chinese balloon, images of the payload, etc.