[perf] Fix the rest UI thread heavy tasks in Brave News on UI thread

[atuchin-m]

A issue to fix the rest of https://github.com/brave/brave-browser/issues/35695 The #perf thread: https://bravesoftware.slack.com/archives/C3T9S9WKD/p1706889037498549?thread_ts=1706626988.579219&cid=C3T9S9WKD

1. large data are still stored/iterated on UI thread: https://github.com/brave/brave-core/pull/21833#discussion_r1477747653
2. other similar places in brave_news that process base::Value and builds something after (e.g. getting the list of sources)
3. private_cdn_request_helper callbacks: https://github.com/brave/brave-core/blob/master/components/brave_news/browser/brave_news_controller.cc#L429-L453

[atuchin-m]

Some extra details: this is ParseCombinedPublisherList work (+the following callbacks) on the UI thread (19ms on a high end workstation)

[MadhaviSeelam]

Verification PASSED using

Brave | 1.70.92 Chromium: 128.0.6613.85 (Official Build) beta (64-bit)
-- | --
Revision | 711f50838f629ac2be4d1d9592f72adecfd3c928
OS | Windows 11 Version 23H2 (Build 22631.4112)

1. Installed 1.70.94
2. launched Brave
3. completed onboarding
4. clicked News >> Turn on Brave News
5. confirmed Brave News feed loaded expected
6. clicked on a news card (eg. Serious Eats)
7. verified news article opened in a new tab (https://www.seriouseats.com/how-to-store-raspberries-8703108)
8. confirmed Manage subscriptions button is shown in the toolbar in the article
9. clicked Manage subscriptions in the toolbar
10. clicked Follow button
11. returned to Brave news feed tab
12. confirmed the publication Serious Eats is shown in Publishers list
13. confirmed New content available. Reload? button shown
14. clicked the button and news feed refreshed
15. clicked Following in the left side panel
16. confirmed Serious Eats cards are shown in the Following
17. return to the news article tab
18. clicked again Manage subscriptions in the toolbar
19. clicked Unfollow button
20. return to news feed
21. confirmed Serious Eats source is removed from Publishers list
22. clicked New content available, Reload?
23. confirmed page news feed refreshed and Serious Eats source was removed from Following as well
24. clicked + sign next to Channels in the Panel left of the feed
25. added a channel (eg.Cars) in the Brave News dashboard
26. closed the dashboard
27. confirmed New content available, Reload?message shown
28. confirmed Cars is shown in the Channels
29. clicked Cars
30. confirmed Channel feed for Cars is loaded

step 5-6	step 7-10	step 12-13	step 15-16	step 17-19	step 21	step 23	step 25	step 27	step 30