prefetch: add another dwc to check the search provider

Revert the changes in I94dbdce72579b545d315353b1cd1de36387faf24
Move the DWC to check the search engine provider

Bug: 345275145
Change-Id: I25f6108dc2f8a556435d2cec19c4123742c1dc56
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5612911


Reviewed-by: Kouhei Ueno <kouhei@chromium.org>
Commit-Queue: Lingqi Chi <lingqi@chromium.org>
Reviewed-by: Hiroki Nakagawa <nhiroki@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1312601}