... excellent exposition!

Thank you very much!

... the empty regex acts as a stand-in for the regex most recently matched (whether the match was successful or not).

In looking for empty pattern documentation (see below), I discovered this is not the case: "If the PATTERN evaluates to the empty string, the last successfully matched regular expression is used instead." (Strange what you can find when you actually read the docs!) Fixed my reply: thanks!

Is this documented anywhere?

The only place I've seen it is in perlop in the Regexp Quote-Like Operators section: the discussion of the  m// operator has a sub-section titled "The empty pattern //" (there's also a brief back-reference to it in the discussion of the  s/// operator).

Are there any typical use cases for employing the empty regex to mean “repeat the regex used in the previous match”?

My vague impression is this is something that evolved early-on as an emulation of shell usage or maybe from a desire for some kind of command line one-liner short-cut facility: saves typing, y'know. Offhand, I can't come up with a compelling example.