Alright, I found an answer that's very clear on the other site. I don't know how accurate it is, but it certainly comes across as knowledgeable.

Both America's Jordan Chiles and Romania's Sabrina Maneca-Voinea challenged the judges' assessment of their routine's difficulty score. America was (initially) successful, Romania was not. Romania did not challenge the fact that they went out of bounds. This is why America's score was adjusted up by 0.1.

Ana Bărbosu, also of Romania, scored equal to Voinea on 13.70, but took the lead due to the tiebreaking rules. Chiles was only slightly lower, on 13.66.

Someone called "Nadia", who is not described in that article but who from my other Googling seems to be "the honorary president of the Romanian Gymnastics Federation", tried to challenge the out-of-bounds call, but because she was not officially Voinea's judge, was rebuked (though bizarrely, the judges apparently did tell her they had proof, rather than just flat-out refusing to discuss it with someone not entitled to have that discussion).

Romania challenged the judges' on-the-ground decision about Chiles' timing and about Voinea's out-of-bounds to CAS, which overturned the initial decision of Chiles' score based on the 1-minute rule, and said that Voinea's timing alsowould have had to happen within the time limit, and because they didn't (Nadia's appeal not counting officially), CAS could not overturn the original call. There was no CAS appeal about Voinea's difficulty score.

Also, @[email protected]' point about most people having plenty of time to lodge a protest, but 1 minute for the last to go does indeed seem to be (at least implicitly) confirmed here.

So the final result:

Ana Bărbosu (Romania) scores 13.70, and there were no questions that her score was entirely correct. She gets bronze.

Sabrina Maneca-Voinea (Romania) scores 13.70. She officially appealed the difficulty but was overturned by ground judges and didn't take it any further. She fails to properly appeal an out-of-bounds penalty and thus her score is 0.1 less than it perhaps should have been. She gets 4th.

Jordan Chiles (America) scores 13.66. She appealed the difficulty and initially had it increased such that her total score was 0.1 higher, but that increase was overturned after a ruling that the appeal came too late. She gets 5th.

As it is, everything actually ends up being exactly the same as the initial call the judges made. 2 people were arguably ripped off by poorly-applied or overly-strict rules, and the one who got bronze actually would have come last (out of these 3) if the loosest application of the rules applied.

All of this basically reinforces my pre-existing belief that subjectively-judged sports like gymnastics, diving, and BMX freestyle are not appropriate for the Olympics.