No, it doesn't, even if your argument requires that it does.

Yes, some folks support structures that make rent-seeking easy while others oppose. In both groups, you have folks who want govt to provide a particular good and some folks opposed. All of thse are still responsible for the resulting rent-seeking, even though clearly they're not "unitary".

> I grow tired of explaining the obvious.

You haven't explained anything. You've dropped a buzz-word that is only tangentially related and made some false assumptions.

I like public choice theory, but it isn't a complete explanation of everything in govt. In particular, while it talks about treating govt as an economic problem, it makes some assumptions about govt that aren't necessary.