I also grew up in the US and came away with the exact opposite impression. Every year from when I was in grade school through university there was a massive push to get everyone to have the flu shot. I didn't hear the phrase "flu vaccine" but the reasoning was always exactly the same as for the covid vaccines: herd immunity, less chance of getting the flu, less chance of dying from the flu.
Now as an adult with no health issues, I go get my flu shot every year. Why not? It's free, it takes barely any time out of my day, and I'm less likely to die.