[wip] Cache bookmark-folder tree and use for system menu

[ayumi]

Related: #4848

Submitter Checklist:

• Submitted a ticket for my issue if one did not already exist.
• Used Github auto-closing keywords in the commit message.
• Added/updated tests for this change (for new code or code which already has tests).
• Ran git rebase -i to squash commits (if needed).

Test Plan:

Reviewer Checklist:

Tests

• Adequate test coverage exists to prevent regressions
• Tests should be independent and work correctly when run individually or as a suite ref
• New files have MPL2 license header

[NejcZdovc]

@ayumi are you still working on this PR?

[ayumi]

I think this is covered by the sites refactor.