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

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 and a bit of
extra code modernization in DeviceCapabilitiesImpl. Based on
https://crrev.com/1511773.

Bug: 443929217
Change-Id: Iec96f2220779c26256e42852c29a2b03da420eb6
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6929105
Reviewed-by: Shawn Quereshi <shawnq@google.com>
Commit-Queue: Lei Zhang <thestig@chromium.org>
Reviewed-by: Simeon Anfinrud <sanfin@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1515567}