So, in the OP play, F6 tags R2 going to third for the third non-force out, and R3 crossed home plate beforehand. But then F6 throws the ball to F3 to play on the BR who is still heading to first but slowing down because he saw the tag of R2.

How is that considered an advantageous fourth out appeal? The BR has done nothing wrong, to that point, to warrant an appeal. He hasn't missed first base. Heck, he's still on his way to the bag. Are you suggesting that a proper appeal can be made while a BR is still heading to the bag?

Or are we now expanding the recognition of advantageous fourth outs to include non-appealable situations where the runner is still on his way to a base after the third out?

The BRD quoted three rules gurus in 2001 that said the BR can be played on for an advantageous fourth out. This can only happen with R2, R3. The ruling has stayed in the BRD to at least 2011. I don't know what the latest JR has.
The WUM does not agree and hopefully it will not occur in your or my games.

It's not an appeal. Those that think so need to go re-read the appeals section. Simply throwing out a runner who has not reached the base he's forced to is not an appeal.

An appeal must be made for some sort of transgression... missed base, leaving early on a caught ball, etc.

Bingo.

Then rule 5.07 is fundamentally flawed and requires a cited Exception.

7.10(d) is also flawed in that it doesn't recognize fourth outs as routine outs. It only recognizes them as appeals for running infractions that occur before the third out is recorded.

Neither is flawed if you agree with the WUM interp, which I'm now inclined to do, having been misled by the BRD interp these past few years which even the author found "non-intuitive."

It could hypothetically happen any time there's two or more runners.

With two outs maybe R3, R1 stealing on the pitch. With one out bases loaded, DP ball to 2B and then somehow R2 gets tagged and BR hasn't made it to 1B. I think R2 and BR getting injured at the crack of the bat would work.