[Brave News]: Add sync support

[fallaciousreasoning]

Resolves brave/brave-browser#23054
Closes #37013
Closes brave/go-sync#429

I had a play around with more complicated sync strategies here but merging gets a lot more complicated unless we add timestamps to individual entires. I think for the most part this should be a pretty massive improvement.

One case where this won't work properly is if you make edits on two different devices, the first edit will win.

[fallaciousreasoning]

Note for reviewers: This is my first syncable pref, so if something looks wrong it probably is.

[github-actions]

📋 Code Owners Summary

4 file(s) changed, 1 with assigned owners

1 team(s) affected: @brave/sync-reviewers

---

Owners and Their Files

@brave/sync-reviewers — 1 file(s)

• chromium_src/chrome/browser/sync/prefs/chrome_syncable_prefs_database.cc

[brave-builds]

Warning

You have got a presubmit warning. Please address it if possible.

Discovered possible removal of preference registrations.

Please make sure to properly deprecate preferences by clearing their
value for a couple of milestones before finally removing the code.
Otherwise data may stay in the preferences files forever. See
Migrate*Prefs() in chrome/browser/prefs/browser_prefs.cc and
chrome/browser/prefs/README.md for examples.
This may be a false positive warning (e.g. if you move preference
registrations to a different place or if the preference was only
written to by the policy stack).

Items:

components/brave_news/common/pref_names.cc: -  registry->RegisterBooleanPref(kShouldShowToolbarButton, true);
components/brave_news/common/pref_names.cc: -  registry->RegisterBooleanPref(kNewTabPageShowToday,
components/brave_news/common/pref_names.cc: -  registry->RegisterBooleanPref(kBraveNewsOptedIn, false);
components/brave_news/common/pref_names.cc: -  registry->RegisterDictionaryPref(kBraveNewsSources);
components/brave_news/common/pref_names.cc: -  registry->RegisterDictionaryPref(kBraveNewsChannels);
components/brave_news/common/pref_names.cc: -  registry->RegisterDictionaryPref(kBraveNewsDirectFeeds);
components/brave_news/common/pref_names.cc: -  registry->RegisterBooleanPref(kBraveNewsOpenArticlesInNewTab, true);

[netzenbot]

nit: pref_registry_syncable.h is only included in pref_names.cc, so this allowlist entry belongs in specific_include_rules keyed to pref_names\.cc rather than the broad include_rules. See BS-030.

[aguscruiz]

HUGEEEEEE

[fallaciousreasoning]

I'm a bit worried about this on initial sync

[github-actions]

Chromium major version is behind target branch (149.0.7827.59 vs 150.0.7871.13). Please rebase.