Potential philosophical issues notwithstanding, it is commonly said that the definition of an elementary particle is an irreducible, unitary representation of the Poincaré group (times a gauge group like $U(1)\times SU(2)\times SU(3)$, of course, but for simplicity I'll just leave that out for the moment).
A key feature of this lies in the fact that this is the universal cover of the group of isometries of Minkowski spacetime, and is in particular a double cover of said group -- this is what allows us to describe the particle's spin in terms of irreps. All this being said...
My Question: Imagine we find ourselves living in a universe where we have a good understanding of how particles/fields/whatever can be described in terms of representations of the symmetry group of the underlying spacetime we inhabit, but we have yet to learn about something like spin. Suppose we have some preferred choice of spacetime, with an associated isometry group that is not simply connected, and as far as we can tell thus far at that this point in history the irreps of this group seems to be fine and dandy for describing the universe around us. Is there some sort of physical reason/insight that might compel us to say, "Hey! Maybe we should look into irreps of this universal cover thingy... perhaps that will reveal something new about nature!"?
In other words, why irreps of the universal cover? Do we just use them because they happen to describe something we know about, or is there reason to suspect they should hold crucial information about the universe?
Granted, this question may be silly and quite likely doesn't really have an answer... but I'm curious nonetheless to have a nice, convincing way of working with irreps of the universal cover other than "it works."
EDIT: I suppose I should say at the get-go that this question is definitely verging on being a duplicate of this one. I will just preemptively say that I think that hinges more on one already having the ideas of quantum mechanics in play (in particular the need for projective representations to deal with the fact that states are only unique up to phase) -- I am asking a bit more generally, even if one knows nothing about quantum mechanics, why they might be lead to consider the universal cover rather than the group itself.