What do you mean by “run”? A gouvernment should fund universities and research institutes who conduct such experiments. I am not opposed to the government having a say in which lines of research to fund in priority depending on what their political project. For example some grant dedicated to energy sources more carbon friendly. But the government should not be more specific topic-wise or method-wise, and should instead rely on panels of experts for questions such as which line of research looks promising enough to put in more gov money.

It’s not a bad idea anytime the government can’t afford to provide whatever service it is they are providing to everyone. If you RCT it, you can then prove how effective it is which can be used to argue for more funding as you can prove the return on investment through improved outcomes.