Yes, the vaccine producers are indeed supposed to prove the negative. They are supposed to prove it will not have such an effect.
Now they have no incentive. They are protected from lawsuits.
As are police officers, in sane jurisdictions, unless they knowingly cause unwarranted harm.
If we removed that protection from police officer that have to make split second decisions to protect the common good, how is it much different from a similar standard of protection for vaccine makers that have to complete many requirements to even get EUA, let alone formal approval for the common good.
To each their own, get a shot, don’t. Don’t care - herd immunity is accomplished in a variety of ways. But calling out an established protection as new to the scene is fake news...