[iOS][Waffle] Update edit button when reloading the settings

Each time the search engine list is updated, the status of the edit
button needs to be updated.

This fix needs an EGTest: b/334780496

Fixed: 334772100
Change-Id: Idbebe99b1f60d591c2ac65ef9ee878d3d589f29b
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5452877
Reviewed-by: Samar Chehade-Lepleux <samarchehade@google.com>
Commit-Queue: Samar Chehade-Lepleux <samarchehade@google.com>
Auto-Submit: Jérôme Lebel <jlebel@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1287284}