From a purely logical perspective, sure. But the rhetorical effect of this statement is clear. Especially, when it contains a lie about the medical data.
"I support safe vaccines, it just so happens that I don't believe that any vaccine is safe and I am not willing to trust the medical system to evaluate vaccines for safety and will continue to insist that all vaccines are unsafe" has the net effect of not supporting vaccination at all.
"I support safe vaccines, it just so happens that I don't believe that any vaccine is safe and I am not willing to trust the medical system to evaluate vaccines for safety and will continue to insist that all vaccines are unsafe" has the net effect of not supporting vaccination at all.