Reland "[power bookmarks] Hook up sync to power_bookmarks"

This is a reland of commit 6bda7fd131b17e0a70244c9c075c0573a5f96f85

It turns out the sync_integration_tests execute a sequence where:
1. PowerBookmarkService is destroyed.
2. Sync makes some call to PowerBookmarkBackend which triggers
   NotifyPowersChanged.
3. PowerBookmarkBackend is destroyed.

At step 2 the observer is a dangling pointer. The fix was passing a
WeakPtr through to the backend instead of a raw_ptr. Also cleaning
up how destruction of the backend_ is done.

Original change's description:
> [power bookmarks] Hook up sync to power_bookmarks
>
> Bug: 1376612
> Change-Id: I6acce22b80560208c8caa2fffd2c1edce23b0ffb
> Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4175972
> Reviewed-by: Rushan Suleymanov <[email protected]>
> Commit-Queue: Brandon Wylie <[email protected]>
> Reviewed-by: Sky Malice <[email protected]>
> Cr-Commit-Position: refs/heads/main@{#1097005}

Bug: 1376612
Change-Id: I3546857c7fa0fa03cae1fd126d2f726ebf6520b5
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/4195156
Reviewed-by: Rushan Suleymanov <[email protected]>
Reviewed-by: Yuheng Huang <[email protected]>
Commit-Queue: Brandon Wylie <[email protected]>
Cr-Commit-Position: refs/heads/main@{#1097613}
27 files changed
tree: ecc0ee7eb3bf1f1b267bc39ebfd854fc7408d4e6
  1. android_webview/
  2. apps/
  3. ash/
  4. base/
  5. build/
  6. build_overrides/
  7. buildtools/
  8. cc/
  9. chrome/
  10. chromecast/
  11. chromeos/
  12. codelabs/
  13. components/
  14. content/
  15. courgette/
  16. crypto/
  17. dbus/
  18. device/
  19. docs/
  20. extensions/
  21. fuchsia_web/
  22. gin/
  23. google_apis/
  24. google_update/
  25. gpu/
  26. headless/
  27. infra/
  28. ios/
  29. ipc/
  30. media/
  31. mojo/
  32. native_client_sdk/
  33. net/
  34. pdf/
  35. ppapi/
  36. printing/
  37. remoting/
  38. rlz/
  39. sandbox/
  40. services/
  41. skia/
  42. sql/
  43. storage/
  44. styleguide/
  45. testing/
  46. third_party/
  47. tools/
  48. ui/
  49. url/
  50. weblayer/
  51. .clang-format
  52. .clang-tidy
  53. .eslintrc.js
  54. .git-blame-ignore-revs
  55. .gitattributes
  56. .gitignore
  57. .gn
  58. .mailmap
  59. .rustfmt.toml
  60. .vpython3
  61. .yapfignore
  62. ATL_OWNERS
  63. AUTHORS
  64. BUILD.gn
  65. CODE_OF_CONDUCT.md
  66. codereview.settings
  67. DEPS
  68. DIR_METADATA
  69. LICENSE
  70. LICENSE.chromium_os
  71. OWNERS
  72. PRESUBMIT.py
  73. PRESUBMIT_test.py
  74. PRESUBMIT_test_mocks.py
  75. README.md
  76. WATCHLISTS
README.md

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