Switch to simpler base::WriteFile() variants (39/N)

This CL is part of a batch of CLs to replace the use of the less
friendly base::WriteFile call with simpler variants, which are easier to
read, and less prone to mistakes.

Changes in this particular CL address files under the path
/chrome/browser/ui.

This CL was uploaded by git cl split.

R=tapted@chromium.org

AX-Relnotes: n/a.
Bug: 418837
Change-Id: Icc8ba4f41fcf6292f36a3ed469ab3009792c2085
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4285652
Reviewed-by: Mike Wasserman <msw@chromium.org>
Auto-Submit: Claudio DeSouza <cdesouza@igalia.com>
Commit-Queue: Claudio DeSouza <cdesouza@igalia.com>
Cr-Commit-Position: refs/heads/main@{#1109313}