If two independent teams recreate this song based on our best knowledge about how the song sounded, can you tell these renditions are two versions of the same song?

Not a perfect test (experts can recognize more similarities) but generally makes it possible to discern "we know jack shit" from "we know the melody, just maybe not exactly how the instrument sounded".

I don't know at which level of detail this line of thinking stops, but it's certainly somewhere after "we don't know the pitch or rhythm for any of the notes in this piece, we can only make educated guesses." Just because we don't have a clear line where the pedantry starts doesn't mean every discussion on the topic is pedantic.

I agree with the other comments here - you should be able to recognize the different renditions as such with an untrained ear.

Basically I’d start with knowing the notes of the song… kinda the most important thing but we just just don’t for this

I’ll note here too that the article is what draws that line (known vs unknown) not me - I’m just disagreeing with where they have drawn the line since there are older songs that we… “know equally little about”

If, presented the original work, I could realistically recognize it in the reconstruction, you're at a good level of detail. Here, we're talking about work in an artform that is embodied by pitch and rhythm, and the "reconstruction" was created knowing neither, so it is very unlikely that it would resemble the original work.