This is an area I am researching at my own pace, general rotations in 3D. I've known about the plate trick for a while as well, and have a very rough understanding of the concept of simple-connectedness and where it may be important.
I'm trying to understand where the importance is with having the ability to spatially orient an abstract entity without 'tangling', in the context of having a topology free from 'handles'.
I can appreciate the utility with this, if there was a need to describe an infinitude of connected, orientable points free from isolated spaces. I'm at a loss to picture where this could be utilized and for what reason, above some other approach.
For instance, fermions are spinors of a sort, so that tells me that a fermion field is such an infinitude of simply-connected orientations and magnitudes in space. Assuming this approach is on the right track, I'm at a loss to see the necessity in the connectedness of orientations. More bluntly, since space-time does not appear to be 'foamy' or 'grainy', I'm not sure how or why it needs to be connected.
Is it as simple as, to put crudely, a requirement as to be able to define what is and is not 'local' to a point in space(time)? If so why does orientation matter or even affect connectedness?