Any use of "force" in modeling physics can be equivalently expressed via conservation of linear momentum
I don't see how. Either you're misunderstanding something, or you have a higher background in quantum mechanics than I do (I've had one in-depth class, and I've read the quantum physics sequence), and it works out like this for reasons I do not currently understand. Which is it?
In any case, force is clearly defined in the simplified version of quantum physics I've learned. It's the gradient of potential energy, which must be specified in the Schroedinger equation. The Pauli principle is not a force. It may be that force is always due to symmetry, in which case calling the Pauli principle a symmetry doesn't separate them at all, but the Pauli principle is still not a force.
The Pauli principle is not a force in the sense that gravity is not force. Yes, you can distinguish between a "force" and the phenomenon responsible for the force (gravity vs gravitational force). What is the difference between these two statements?
1) That's not a force, it's the playing out of the fundamental symmetries in quantum physics, normally phrased here as the Pauli exclusion principle.
2) There's no force on that falling object in a vacuum, it's just following the geodesic dictated by the symmetries in General Relativity.
I thought this video was a really good question dissolving by Richard Feynman. But it's in 240p! Nobody likes watching 240p videos. So I transcribed it. (Edit: That was in jest. The real reasons are because I thought I could get more exposure this way, and because a lot of people appreciate transcripts. Also, Paul Graham speculates that the written word is universally superior than the spoken word for the purpose of ideas.) I was going to post it as a rationality quote, but the transcript was sufficiently long that I think it warrants a discussion post instead.
Here you go: