proves that the ideas around it were very much alive
Do you have sources? I do not see any sources which ground this. Except for the Jews. 637 Arabs were Jews then? Then why they prevent Jews to build?
In such a situation taking the counter position and building something is not a big step but a logical step to make a rebellious point.
It is your deduction. Do you have sources to ground it?
(For me ...) Arabs who have built did not did it because Christians said the contrary: "One does not want a building there!" "Guys! it is because you do not want that that we want to build one !!!"
I do not think 637 Arabs as childish people.
It would have been different if no one in 7th C remembered the fuzz about the Temple Mount.
The Jews not remembering the Temple? You're serious?
Then a mysterious author who dug that up out of ancient texts could have introduced it out of the blue.
Plausibly in Q 2:125-27.
But with what we know, the temple Mount was a living issue at the time.
Yes, for the Jews, no one else. (Unless you bring new sources...)
The Quranic author could have picked up the already living sentiment and have written it down.
Unless you have (new) sources, the "living sentiment" is Jewish. Then why Arabs did not let the Jews build? Because they did not have a clue about those texts they have, apart the fact that it was written in Arabic and with that simple fact it could not concern the Jews.
Whom then?
Them.
The Quran is not necessarily THE inspiration of the 637 Arab action to start building. imo.
I do not say "necessarily". I say plausibly. I do not see where this idea would have came from.