I don't think anyone is proposing that government employees (or politicians) actually design the rockets themselves. The question is, can governments oversee and fund the development of innovations like VTOL rockets? And the answer to that question is "yes"[0].
As for why innovation in that field stopped, and had to be picked up again by private companies, I think it's worth asking whether large companies have undue influence over the sorts of projects that governments choose to pursue.
What? That makes no sense. We already had a reusable vehicle for space flight. We've had it for decades now. It's called the Space Shuttle. NASA developed it in 1981. Just support NASA and science for the public good. Do you really want a handful of billionaires in control of space exploration? They do it for their own benefit not ours.
Government is good when the public externalities are large, the private benefit is small and there's an extremely large capital requirements.
This is not the case any longer for rocket tech, and so it makes zero sense to have the government do it. It'd be like asking the government to make smartphones, which is an obviously ridiculous proposition.