If it's been proven that anything has potential to be politicized, why do we speak of so many ideas as being outside the left/right spectrum, even when they actually influence a left/right philosophy?

I'm helping to build an instance, and we had a debate recently on whether Platonism is left-winged or right-winged. It's an ancient philosophy, mind you. Created by Plato. One small side said it was right-winged, one small side said it was left-winged, and the majority said it couldn't be either. Someone remarked "what do you mean it's neither? Marx cited him!" Admittedly it's frustrating when you're researching these things so you can give it a respectful comparative review and someone says "you can't judge people of ancient times based on your left-right mode of measurement" in a world where something like whether Obama can do a public prank April Fool's Day is a "political" issue (remember when he said he was building Iron Man as an April Fool's joke and everyone on the right claimed it was unprofessional while the left enjoyed his sense of humor). That's somehow more worthy to put under the microscope than Plato, the world's first "public" philosopher (after Socrates and Thales who weren't of specific opinions and Ptahhotep who was more of a superior advocating an approach that worked for him)?

In my eyes at this point, as well as the eyes of the groups I help out in, everything is equally politicized as a default; that is, "politicization" is what the individual makes of it at a given moment. But I know that isn't how the world operates. Marx himself was known to write about an enormous number of topics, from faraway cultures to appropriate punishments for oddly specific crimes. How does the inherent potential of everything that exists to be politicized square with the idea that certain things are also inherently seen as non-left-or-right based on the circumstances that they hold in their own setting?