Conservatism, practically by definition, resists and rejects new things.
Conservatives reject blindly accepting new things, yes. That's called looking before you leap and taking time to rationally weight the risks vs rewards.
Society is a complex machine. It's impossible to gauge the effects of something like a vaccine in a few short lived lab tests. If we inject a vaccine into children, during their most formative years, how is that going to effect their development a decade from now? Is it going to stunt their growth? Increase their risk of cancer? Give them some other subtle illness years down the road? And for what benefit? All scientific evidence so far says that children have essentially no risk of being injured by Covid.
But asking such reasonable scientific questions gets you branded anti-vax and anti-science by people who understand neither.
We know the elderly with pre-existing conditions do have a substantial risk of being injured by Covid, and even if the vaccine has long-term health effects, that's not something someone with an average 10 years of life left has to worry about. So that cost/benefit analysis makes sense for them to take the risk on a new vaccine.
Progressives are prone to blindly accept anything new for novelty's sake, especially from government, which they use as a substitute religion. No one screaming for everyone to take the vaccine is doing so because they're a scientist and have read peer-reviewed studies showing it's safe. They're doing it because it's a change, which they love, ordered by their government masters, which they irrationally believe unconditionally.
You're making several false assumptions. The working class, who are largely young and healthy, aren't dying in any meaningful way from not taking the vaccine. They're more likely to die in a car crash or from heart disease, than from Covid.