Remove some deprecated base::JSONWriter::Write() calls in various dirs

Mechanically switch from base::JSONWriter::Write() to base::WriteJson()
for callers that did not check the return value. Keep the existing
behavior and return an empty string on failure.

This CL was initially AI generated, with human touch-ups. Based on
https://crrev.com/1511773.

Bug: 443929217
Change-Id: I35762c03663c85d8a291ca2f1bd638e0ceadd328
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6929523
Reviewed-by: Kyle Charbonneau <kylechar@chromium.org>
Owners-Override: Kyle Charbonneau <kylechar@chromium.org>
Commit-Queue: Lei Zhang <thestig@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1513868}
NOKEYCHECK=True
GitOrigin-RevId: d42cf56c54db361576eaea45c91077a418e53f0a