"I think some of the live virus vaccines are probably averting more problems than they’re causing. There’s no vaccine that is, you know, safe and effective."
N.B. from a logic perspective there's no conflict between "I would like this tech to work well" and "I don't think it works well [enough]" as positions, they're not mutually exclusive. Additionally, words like "safe" are relative. It's possible for a vaccine to avert more problems than they cause, whilst still creating sufficient damage that a reasonable person would consider them to be unsafe.
He then goes on to talk about the history of US vaccine legislation, which states that vaccine manufacturers are given blanket immunity for vaccines when they are "unavoidably unsafe", and that every vaccine is treated as immune under this rule.
This is a common point of confusion in discussion of vaccines. Most people make an ambient unstated assumption that vaccines are medical products and must therefore be safe, because otherwise they wouldn't be licensed. In fact US law considers them to products that inevitably by design make some healthy people sick or permanently disabled, and exempts them from the usual product safety rules as a consequence.
That RFK believes these things doesn't mean he wants to eliminate vaccines. You could at best argue for this indirectly, by arguing that he wants safety standards raised (put back) to the level that other treatments are held to, which would have the effect in practice of making vaccines unsellable, and thus that by implication he wants this outcome. But that is a tricky argument to make.