Does playing LeBlanc Everywhere at Once in response to the temporary trigger to a battlefield prevent the temporary units from dying?

No. Playing LeBlanc, Everywhere at Once after the Temporary trigger is already on the chain will not stop those units from dying.

Here's why: Temporary is a triggered ability (Rule 816). When the Beginning Phase starts, it triggers and is placed on the chain as a pending item. LeBlanc's passive says "Your [Temporary] effects at my battlefield don't trigger" — this prevents the triggering event, but once the trigger has already fired and is sitting on the chain, her passive does not remove it. The trigger resolves normally and kills the unit.

This is the same principle as FAQ #11281 — reacting to a Temporary trigger doesn't stop the original unit from dying.

The scenario where LeBlanc does help is if she's already on the battlefield before the Beginning Phase starts (FAQ #10040). In that case, the Temporary effects never trigger in the first place. Playing her afterwards is too late for this turn. She will prevent future Temporary triggers on subsequent turns if she remains there.