For every person who is skeptical to the "new" vaccines you'll find a person skeptical to the "old" vaccines, and I'd be surprised if there wasn't significant overlap (i.e. the people saying that they don't want to take the COVID vaccines because they're too new suddenly saying that they don't want to take the smallpox vaccine for $reasons).

Doesn't help that the "standard" smallpox vaccine is quite nasty in terms of application and side effects (leaves a scar): https://en.wikipedia.org/wiki/Smallpox_vaccine#ACAM2000