I really do not understand why the perl context cannot be retrieved from TLS when it is needed, rather than passed around like a baton?
Because on linux pthreads at least, retrieving a value from TLS is very expensive (using the "match the SP to a thread" trick). Passing an extra parameter is a lot quicker.
As to why they don't reserve a register for it, you'd have to ask them. But I would speculate that it boils down to the fact that you can't assume that all code from all libraries compiled by all the different possible compilers on all the different platforms all know and agree that they're not allowed to touch register X. If this had been agreed on day 1, then maybe; not so much now.