Well.
The flu isn't always mild, especially as you get older. Depending on the year a couple hundred thousand Americans wind up in the hospital and about 10-20k die. [1] Compared to COVID-19 (hence the "politicization of getting the vaccine) it's not that bad but it's still not great.
Now what is the cost of getting a flu vaccine every year? Well, for most people it's 0. You walk in, get a shot, walk out. Arm is sore for a day or two at most and that's it. Then in getting the flu shot you reduce your own chance of getting the flu and reducing severity if you do get it anyway (which many people do), and you lower the risk for others who might be immunocompromised, susceptible to the flu, etc. Now I understand if you want to make the case that society shouldn't pay for it or that it's not worth it or w/e, but on an individual level there seems to be a ton of upside versus the downside.
So what's the downside of the flu shot? You're inconvenienced. That's about it. Unfortunately it's hard to measure the value of that inconvenience versus the upside, so, as typical human nature goes, you just value your inconvenience and then decide not to get a flu vaccine. Unfortunately it's just human nature. The same human nature manifests in all sorts of "I got mine" kinds of ways, ranging from global warming to opposing healthcare for all (why don't they just get insurance like me?).
If anything I think the dividing line isn't so much political (though it's now being used politically), it's who is willing to make sacrifices for the good of society. In the case of the flu vaccine here what I'm hearing you say is basically "welp, I don't see how it benefits me so I don't care". Is that not the case?
> it seems deeply silly to me to get a yearly shot just to cut down on the chance of mild illness
As someone who now just grabs the flu shot whenever it's available and I happen to be in a convenient spot the 1-3 days I might feel like crap seem like an awfully high price to pay for not getting a shot that takes 5 minutes. At least practically speaking. Idk. I don't like getting sick and the shot doesn't bother me at all.
[1] https://www.cdc.gov/flu/about/burden/index.html