The important thing is that these things get funded. It doesn't matter what institute funds them. If an institute becomes stultified and corrupt, there's no reason to champion it over creating another.
That's true, but I have lots of experience with the NIH, and haven't found it stultified nor corrupted. In fact, did you hear they recently funded a project that reversed type 1 diabetes?
Besides what others have said, the government is immune from many of the multi-agent coordination problems that trap other types of entities. It's basically the essential reason we have government at all.
So no, not all things government does can be replaced by the private sector, for reasons of game theory.