The implication of that being the shootdown was a media-initiated event. After civilians saw the balloon from the ground, and it became lead story on every news site, with the MAGA press calling Biden "weak", something had to be done, whether it makes military sense or not.
I mean it's game theory. Once you start allowing such things you simply have to always allow them. If you allow China to bully you into not visiting Taiwan, they will start bullying you into something else. Neither side will actually do war so it makes sense not to fold
Ah yes, but like others have pointed out, China did in fact allow US ballons over their territory.
(googles loon project)
"Neither side will actually do war so it makes sense not to fold"
And neither side wants war with each other, but if either side is pushed more and more into it from internal and external pressure, than I sadly would not rule out that possibility.
USA is being baited into war on multiple fronts.