"It's fairly easy to design something and show that it's safe." To spend 2 million man hours and 1/2 a billion dollars on creating documentation for certification? I would agree that one never knows if something will be successful until it is, but if it was "fairly easy" to get approval to build working reactors then a lot more people would be building them. I think a way to get a lot more innovation in this space would be to designate an area in the US, maybe the former nuclear testing area, as a place where the radiation protection rules could be relaxed and work on reactors could be done at a much quicker pace.
The problem is not with radiation. It's cost. There's no efficient way of "trying out" a design. Even if we would allow anyone to build almost anything with constant NRC monitoring, the cost would kill the project. (And we wouldn't really gain much insight, because of complexity. This is basically the same problem that we have with the FDA. Yes phase I-II-III clinical trials are expensive, but biology is complicated.)
Now, that said, the whole problem is that of scale. There's no real money in building nuclear power plants. It's basically a few fanatics doing it at new and old companies.
If the US would announce a 2T USD new plant budget, there would be competition.
And even then, probably we would need to build a few very similar plants to have some sort of efficient design.
