Run Scores Or Not
 

Do you agree with my wording of this case?

Runners on 1st and 3rd with 1 out. BR out on outfield fly ball. Defense appeals R2 left 1st before the catch and R2 is called out on the live ball appeal, the third out. R1 tagged up properly and scores.
Because BR was out before the appeal, it is not a force out, so run(s) can score subject to timing. If a live ball appeal, the lead runner must reach home before the appeal.

USA SB Rule 5.5.B.1

I am not too sure about that.
The way I read it, 5.5.B.1 applies to a force out. Even though it mentions the word "appeal", the context seems to be an appeal related specifically to a force out, which you do not have in your sitch.

I can certainly be wrong, but I think the rule that would apply is 5.5.B.3 "A preceding runner is declared out on an appeal play"

The way I am interpreting this, is that no run shall score regardless of the timing of the appeal on a preceding runner being called for the 3rd out for not tagging up.

The OP case is an example of the run being able to score because R2 is not forced and the timing was R1 reaching the plate before the appeal.

Rule 5.5.B.3 applies to a preceding runner being appealed out.
R1 is not preceded.

I stand corrected on the English language definition of the word "preceding". :o

However, I am still kinda hung up on the wording of the rule you are citing:
"On an appeal play, the force out is determined when the appeal is made, not when the infraction occurred."

There is no force out in the OP.

So, still not sure.

A hard one to explain in words.

The OP has no force as you said.
Under the rule, if the force was still in effect but needed an appeal, the run depends on the timing. If a forced runner trailing the one scoring misses a base, and then is out on appeal; the run timing is based on the time of the appeal (live ball) and not the time the runner should have touched the base.

It is USA SB only, adding to the confusion.

My OP question was about clarity of wording, so Tony helped a lot.