Partial answer.
The reason relativistic theories are so restrictive is because of the rigidity of the the symmetry group. Indeed, the (homogeneous part) of the latter is simple, as opposed to that of non-relativistic systems.
The isometry group of Minkowski spacetime is \begin{equation} \text{Poincare}=\mathrm{ISO}(\mathbb R^{1,d-1})=\mathrm O(1,d-1)\ltimes\mathbb R^d \end{equation} whose homogeneous part is $\mathrm O(1,d-1)$.
On the other hand, the isometry group of Galilean space+time is \begin{equation} \text{Bargmann}=\mathrm{ISO}(\mathbb R^1\times\mathbb R^{d-1})\times\mathrm U(1)=(\mathrm O(d-1)\ltimes\mathbb R^{d-1})\ltimes(\mathrm U(1)\times\mathbb R^1\times\mathbb R^{d-1}) \end{equation} whose homogeneous part is $\mathrm O(d-1)\ltimes\mathbb R^{d-1}$, which is not semi-simple.
There is in fact a classification of all physically admissible symmetry groups (due to Carroll?), which pretty much singles out Poincaré as the only group with the above properties. There are is only one family, which contains two parameters: the AdS radius $\ell$ and the speed of light $c$.
Than being said, note that
Reeh-Schlieder: if $c\to\infty$ all spacetime intervals are spacelike and therefore there is a meaningful notion of absolute time. The locality axiom $[\phi(x),\phi(y)]=0$ for all $(x-y)^2<0$ becomes vacuous, because we have $(x-y)^2<0$ for all $x,y$.
Coleman-Mandula: The rotation group is compact and therefore it admits finite-dimensional unitary representations. This is probably the step where the theorem fails. I'll think about it.
Haag also applies to non-relativistic systems, so it is not a good example of OP's point.
Weinberg-Witten. To begin with, this theorem is about massless particles, so it is not clear what such particles even mean in non-relativistic systems. From the POV of irreducible representations they may be meaningful. But they need not correspond to helicity representations (precisely because the little group of the reference momentum is not simple). Therefore, the theorem breaks down (as it depends crucially on helicity representations).
Spin-statistics. As in Reeh-Schlieder, in non-relativistic systems the locality axiom is vacuous, so it implies no restriction on operators.
CPT is interesting. I'm not sure where exactly the proof of the theorem breaks down if the symmetry group is Bargmann. I'll have to go through the details of the theorem.
Coleman-Gross. Is this result violated in non-relativistic systems?
I have to leave now. I'll think about it and perhaps fix the answer and undelete it later on.