What's the worst that can happen?
I suppose a lot of the covid vaccine side effects are likely to apply to flu as well in principle, but there isn't the same fuss about it because it's not new. It's a bit like all the accounts of mobile phones making people ill, they didn't arise at the time people started using mobile radio, they started when mobile radio became new to the general public rather than just a handful of professional users. If something's new it gets attention, and if it gets attention it becomes the subject of all sorts of post hoc conclusions.
I had my first flu vaccine last January, and I was fine. I'm getting another in the next few days, now that I've had the covid booster. I had quite a violent reaction to a typhoid vaccine once even though the girlfriend didn't, and I suppose that luck of the draw applies to pretty much every vaccine/drug, but they don't get the same publicity. Most drugs have a horrendous list of side effects on the leaflet, including the OTC ones, but nobody pays any attention.