It is not that hard to show, by explicit computation, that a finite-dimensional irrep of a Clifford algebra with $n$ generators $\gamma^{\mu}$ (in the Dirac theory, $n$ is the dimensionality of spacetime) must be $m$-dimensional, where $m=2^{[n/2]}$. This is mostly a matter of showing that the fundamental anticommutation relations imply that the algebra contains at least $m^2$ linearly independent elements. That a $m\times m$ matrix irrep exists is then easy. (Dan Freedman had some good lecture notes on this.) This basically covers the universality for finite-dimensional representations.

Why do you think that the collection of all positive linear functions on $A$ is the actual object of interest?

I don't think there is a rigorous derivation. For one thing, it is possible to construct systems in which the physically-correct time evolution does not correspond to a minimization of the action. It could be a maximum of the action or—more crucially—a saddle point of the action. But what does it mean, precisely speaking, to be a saddle point of the action? Well, the fields obey the Euler-Lagrange equations....

@TimothyChow No, it's not quite the same, you're right. It does seem like a relevant point of comparison though, since it expressed a converse situation where what looks like a "probability 1" situation is not (necessarily) generic.