Note that my sentiment/tone in asking this is "huh, if that's the case then POWER and other architectures could really compete with x86!". (Assuming POWER doesn't use that approach.)
You are describing "register renaming". And it's what everybody discovered was a better idea than putting "specific tasks" to registers with windows, banks, etc.
OoO register renaming is a nearly orthogonal concept compared to SPARC style register windows. And I say nearly, mainly because if you try to implement both it increases the complexity more than you would think due to microarchitectual interactions between the two concepts.
Also, it's not really an x86 thing (for instance most Atoms don't have these extra registers), it's an in-order vs out-of-order thing. Most Power cores do have the larger bank and renaming.
These turned out to be not such a great idea.