While technically one might say that the anecdote is about “spreading out the [existing] languages” rather creating these languages, I don’t think you can make the anecdote carry the structural load in this way.
Yes… we could say there are already the many languages… but if they were all joined together in making a construction … and had to suddenly stop because something changed in their communication…
… this implies that there was a COMMON language in addition to the different languages… and that suddenly the workers could no longer understand the common language.
This is far less plausible than simply going with the original idea:
Everyone spoke one language… and then various groups suddenly had their own language.
How else would you describe it, @swamidass?