That's an interesting example, but it doesn't have anything to do with "putting the physical register file to use". Register renaming always helps as long as your algorithm is dependency-free. It's probably one of the single best general purpose optimizations available to a modern CPU design.

The classic example of where register-poor ISAs hurt isn't about "subtle differences" at all. It's the fact that there are still only 8 (for i386) named registers, and so anything that needs to deal with a working set beyond that needs to do spill/fill to memory, and you can't "rename" memory accesses (though you can sort of cheat, as with the store forward optimization -- but that doesn't work nearly so well as renaming does).