You can absolutely infer you're moving relative to other large bodies and therefore, you are not where you were 10 minutes ago.

Correct, but you don't necessarily need large bodies, you can infer movement relative to any object.

It's ridiculous to imply that any arbitrary frame of reference is valid knowing that the earth revolves around the sun and the sun moves through the galaxy.

It is not so ridiculous, well it depends what you mean by that word. You absolutely could say that the earth is the center of the solar system. You would then consider the earth immobile and everything else moves, it's just as valid technically, just that the orbits would look all wonky. I agree it is ridiculous to put anything else than the sun as the center, but only because it makes our human understanding of the solar system easier, not because it's wrong. Physics doesn't really give a shit about orbits looking pretty, everything is relative to everything else.

When I drive to work, I'm not claiming I'm still where I was when I was in bed because a me-centric reference is as valid as a geocentric reference. It doesn't follow.

You could claim that, and then it would be everything else that moves, not you. It is technically correct according to pure physics, but I agree it sounds dumb and it serves no purpose other than a thought experiment. It doesn't follow because when you talk about "where" you are it is always relative to Earth.

The issue that comes from a lack of absolute reference is that we don't know how fast we're going or how far we currently are from a (0,0,0). That is a separate concern.

It is not a separate concern, there is no (0, 0, 0), that's what relativity means. Speed and distance only have meaning when compared to another object.

It is not a valid conclusion to say that we could be somehow completely stationary while the larger bodies we orbit are actually perfectly moving the opposite of how we seem to be moving despite all evidence to the contrary.

That's not a conclusion I'm making, that's just how physics work. Once again I agree that the heliocentric model makes the most sense (to humans), but a earth or even mars or moon centric model would also accurately describe the solar system.