Reopen sync code dialog when switching desktop <=> mobile

[AlexeyBarabash]

Resolves brave/brave-browser#23262

The behavior which happens in the original issue doesn't leave any traces in console. It acts like click happened not on the View Sync Code link, but somewhere outside of the modal dialog, this line at onPointerdown_(e) {}

This PR forces to re-open the dialog when its type is changed. This introduces flickering, but UI is always responsible.

Submitter Checklist:

• I confirm that no security/privacy review is needed, or that I have requested one
• 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, npm run lint, 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/Custom-Headers
  • 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:

Please see brave/brave-browser#23262

[AlexeyBarabash]

@petemill , PTAL

[petemill]

👍 Good find. Whilst it would be good to solve within the same dialog, I think closing and opening the dialog is completely acceptable since it's an implementation detail that it happens to be the same dialog. It just as well could have been separate dialogs.

[kjozwiak]

Using the verification that @MadhaviSeelam completed via brave/brave-browser#23262 (comment) as the check on Nightly for uplifting into 1.41.x & 1.40.x as per #13812 & #13813.