I specified "widely used", which effectively excludes brand-new vaccines.

I don't have any issue with what you're describing. The J&J COVID vaccine would be an example of caution being advisable, since you never know what unusual interactions (e.g. with blood clotting) might occur in a larger population than the safety studies looked at.

But that's not the usual definition of an anti-vaxxer.

Anti-vaxxers carry on about things like thimerasol which (1) were removed from most vaccines 25 years ago and (2) give you an exposure to mercury equivalent to eating something like a single can of tuna.