That's not exactly how the flu shot works. They make an educated guess as to which strain will become most widespread, then develop the corresponding vaccine. Years that the flu shot isn't as effective is when they guess the wrong strain.
My guess is that next year's flu season is going to be a lot worse than more recent ones because of this.
Whether we do it, though, is another question altogether.
Just putting a mask on half the symptomatic infected people (or even better, getting them to stay at home) would probably do the trick.
Also, the flu has to start from a very low level. Might miss this flu season completely.
Do we have any sign yet on whether the COVID-season would be expected to run on a similar schedule?
(the "spike" that the vaccines target is important to the virus, so changes to it often make it much less infectious or whatever, and we don't really know how much room is has to both escape the vaccine triggered immunity and stay highly infectious)
Doing anything like this systematically should be though out carefully.
Doing my first 10 vaccines felt like buying software licence in the 90'. I fear that the new ones will be like those sucky Saas subscriptions.