SidePanel: fixes button pressed state.

[mkarolin]

Resolves brave/brave-browser#41154

It's a regression that only SidebarContainerView::OnRegistryDestroyig() is called when active tab with ai chat is closed now.
Instead, we could try to make sidebar controller refresh active item state whenever relavant event happens.

Submitter Checklist:

• I confirm that no security/privacy review is needed and no other type of reviews are needed, or that I have requested them
• There is a ticket for my issue
• Used Github auto-closing keywords in the PR description above
• Wrote a good PR/commit description
• Squashed any review feedback or "fixup" commits before merge, so that history is a record of what happened in the repo, not your PR
• Added appropriate labels (QA/Yes or QA/No; release-notes/include or release-notes/exclude; OS/...) to the associated issue
• Checked the PR locally:
  • npm run test -- brave_browser_tests, npm run test -- brave_unit_tests wiki
  • npm run presubmit wiki, npm run gn_check, npm run tslint
• Ran git rebase master (if needed)

Reviewer Checklist:

• A security review is not needed, or a link to one is included in the PR description
• New files have MPL-2.0 license header
• Adequate test coverage exists to prevent regressions
• Major classes, functions and non-trivial code blocks are well-commented
• Changes in component dependencies are properly reflected in gn
• Code follows the style guide
• Test plan is specified in PR before merging

After-merge Checklist:

• The associated issue milestone is set to the smallest version that the
  changes has landed on
• All relevant documentation has been updated, for instance:
  • https://github.com/brave/brave-browser/wiki/Deviations-from-Chromium-(features-we-disable-or-remove)
  • https://github.com/brave/brave-browser/wiki/Proxy-redirected-URLs
  • https://github.com/brave/brave-browser/wiki/Fingerprinting-Protections
  • https://github.com/brave/brave-browser/wiki/Brave%E2%80%99s-Use-of-Referral-Codes
  • https://github.com/brave/brave-browser/wiki/Web-Compatibility-Exceptions-in-Brave
  • https://github.com/brave/brave-browser/wiki/QA-Guide
  • https://github.com/brave/brave-browser/wiki/P3A

Test Plan:

See the linked issue for manual test.
SidebarBrowserTestWithAIChat.TabSpecificPanel

[mkarolin]

cc: @simonhong, please let me know what you think.

[simonhong]

@mkarolin Added my changes. PTAL :)

[mkarolin]

Seems to work well. Thank you @simonhong

[simonhong]

@petemill Kindly ping :)

[simonhong]

SidebarBrowserTestWithAIChat.TabSpecificPanel failed after cr130 bump. checking.
Set as draft as this PR doesn't work with cr130. Fixed.

[brave-builds]

Released in v1.73.6