Remove some deprecated base::JSONWriter::Write() calls in //chromeos

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: I95b879252d55dc608a4b217cbaf956a09e5a4969
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6929025
Reviewed-by: Daniel Cheng <dcheng@chromium.org>
Reviewed-by: Ryo Hashimoto <hashimoto@chromium.org>
Commit-Queue: Lei Zhang <thestig@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1513526}