[Code Health] Remove a use of base::SupportsWeakPtr. (widget)

Bug: 40485134
Change-Id: I06472478fd5b1177914d55d4f910dd1de5db0cc7
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5608335


Reviewed-by: Dave Tapuska <dtapuska@chromium.org>
Commit-Queue: Dave Tapuska <dtapuska@chromium.org>
Auto-Submit: David Bertoni <dbertoni@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1313020}