As someone who stopped getting the flu vaccine 4 years ago because I finally admitted always got sick for at least a week and a half (to the point of having to miss school or, now, work) starting 1-2 days after getting it, every year for 20 years, no it isn’t.
Note: my comment above is not about the covid vaccine, only the flu vaccine. I do hope we start having an mRNA based annual flu vaccine soon, though. And lots of other new and better vaccines.