[OmniboxLabs] Allow post done updates from unscoped extensions provider.

This pretty much follows how history embeddings allows post done
updates.

Change-Id: Idbf0763e8a22099988f99772c428746ae0acb46c
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6204183
Commit-Queue: Nihar Majmudar <niharm@google.com>
Reviewed-by: manuk hovanesian <manukh@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1413585}