Sure it could happen that way but it doesn’t say exactly that it happens that way. The Bible doesn’t say exactly at what point in thee end times that the temple starts to be rebuilt only that it get s rebuilt in the “latter days”.
Perhaps it does get rebuilt over the Dome...perhaps an earthquake takes it out.
We only know that it does get rebuilt. Where that happens is conjecture and a reliance on tradition or traditionalist sentiment.
The Bible doesn’t even say that it is a good thing that the temple gets rebuilt...since sacrifices are restarted which is a rejection of the final sacrifice of Jesus for our sins! Yet we do know that God is terribly displeased when the Beast commits the abomination of desolation there.
Listening for the final trumpet - but those six before that one sure will be loud.