Must conclusions from relativistic physics hold in non-relativistic physics?

Physics is not just a branch of math: it is a method for modeling phenomena in the real world. If a fact is proven experimentally, but a theory fails to account for it, it is a problem with the theory, rather than with the reality.

E.g., spin arises naturally in relativistic theory, but there is no reason why it should exist in non-relativistic quantum mechanics. Yet, we do include the Zeeman term in the Schrödinger equation, since otherwise we wouldn't be able to describe the spin-related phenomena. Same is true for symmmetrizing/antisymmetrizing the wave functions.

It boils down to two things:

1. You're right that in a general nonrealativistic theory, spin-statistics theorem does not necessarily hold.
2. But we assume that our actual nonrelativistic physics is really only an approximation to a more fundamental and relativistic nature. Thus, the nonrelativistic theories we use to describe our world carry an imprint of the underlying relativistic theory.

The relativistic case is a more general description of how the universe operates, and the "non-relativistic" case is a simplified approximation that only works in certain cases.

It's important to note that, contrary to your phrasing of "switch of reference frame", exotic effects such as time dilation or unification of forces aren't something you simply switch on or off—instead, they're phenomena that vary in degree based on (velocity, temperature, gravity, other parameter). We have investigated these phenomena in a variety of conditions, and we have models that appear to reliably describe the behavior of reality.

As an analogy, consider the function $f(x) = \sin x$. We know that this function is a periodic curve, but in the very special case where $x$ is close to zero, the function looks pretty close to linear, and sometimes we can use the approximation $f(x) = \sin x \approx x$. (Undergraduate classical mechanics would be impossible without the magic phrase "for small angle $\theta$"!) Any model that claims to represent the behavior of a sine-based function at large or all angles must, in the special case of $x \approx 0$, produce results very close to the approximation, but that's not because we "switch" our model, it's because the approximated model is intentionally trading a small amount of accuracy for tractability.

So yes, the conclusions from relativistic physics must hold in non-relativistic situations specifically because we are saying that "this is a more complex model that describes the way things work in a wider variety of situations than the simpler model". If the more complicated model does not (after performing a lot more calculations!) produce answers that are so close to the simple everyday model that we can't tell the difference, then its claim to describe the universe fails. (In fact, this is how we know that neither GR nor QM as we currently have them is "correct"—they don't extend into each other's domains.)