It's not always effective, because they have to predict when they make it what strains of flu will be prevalent in the next flu season, and they don't always get that right, but in that case you are no worse off than if you had not gotten the vaccine. But when they do get it right, it can save you from getting the flu. That's certainly worth a few minutes getting it an a couple days with a sore arm.
With a COVID vaccine, I doubt it will get anywhere near the same fraction of takers as the flu vaccine.
1. The anti-vaccine crowd probably won't take it.
2. The "COVID is no worse than a mild flu" crowd probably won't take it. Even if they do, it won't be at a higher rate than they take the flu vaccine.
3. The "COVID is a hoax" crowd probably won't take it.
4. A lot of the people who believe COVID is real and serious will probably be turned off if the approval seems to have involved politicians forcing approval over the objection of scientists who say it is not ready yet. (Especially if those politicians have also been pushing the "no worse than a mild flu" narrative).