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Hello Herman!
It would be useful to know whether this is used somewhere or in discussion.
I don't know if it is used anywhere. It is very similar to double/triple elimination, of course, and is a rather obvious idea, so I decided to write it down.
To me it looks like a reformulation of double elimination (is there a practical difference at all, I don't see one) that would allow to generalize it. Ok, obviously the main value would be the framework for triple, quadruple elimination, because there is a rule for pairing down when necessary.
I see a potential concern with this system, though:
In the double elimination example, the sole undefeated player gets two byes after Round 4 because Round 5 reduces five players to three. As a result, the de facto #2 player would need to win seven games while the #1 seed only needs to win five games. I'm not sure whether the difference between #1 and #2 is so great to justify this discrepancy...
(One possible tweak might be handing out byes in such a way to maximize the chances of getting back to a power of two. In this case, Round 5 will only involve #5 and the lowest seed who hasn't played #5, which would most likely result in a Round 6 involving four players including the #1 seed.)
For this reason, I would favor a direct final, which means either the undefeated player wins with 5/5 (other player has 5/7) or the challenger wins with 6/7 (other player has 4/5), which seems reasonable.
In standard double elimination, the winner bracket does not play in odd rounds (except round 1). In triple elimination, the winners bracket plays even less.