If you take the view that government is needed to protect people's rights (you know, "to secure these rights governments are instituted among men") as opposed to providing people with solutions for their problems - then the question becomes easy. Why should we? Who said scientists are better in protecting one's rights than, say, cops or doctors or lawyers or plumbers? I do not see anything in scientific training that…
False dichotomy.
Of course, the government would always have a lot of purely technical questions where scientific training would allow one to decide them efficiently - but these always would be secondary roles and rarely need to be elected, but rather properly hired, which can be done by any decent manager.
If politicians were people who give difficult problems to a team of scientists to allow them to come up with the best solutions, and then choose from the equally good ones based on the underlying philosophical principles they stand for, then no, we wouldn't need scientists in the government.
Failing that, I'll take the group who at least in principle deals with new problems through a better method than guesswork or trial and error.