refactor(browser-platform-utils): Remove Deprecation and Fix Code

[Patrick-Pimentel-Bitwarden]

🎟️ Tracking

https://bitwarden.atlassian.net/browse/PM-21438

📔 Objective

Removed references to deprecated functions.

📸 Screenshots

⏰ Reminders before review

• Contributor guidelines followed
• All formatters and local linters executed and passed
• Written new unit and / or integration tests where applicable
• Protected functional changes with optionality (feature flags)
• Used internationalization (i18n) for all UI strings
• CI builds passed
• Communicated to DevOps any deployment requirements
• Updated any necessary documentation (Confluence, contributing docs) or informed the documentation team

🦮 Reviewer guidelines

• 👍 (:+1:) or similar for great changes
• 📝 (:memo:) or ℹ️ (:information_source:) for notes or general info
• ❓ (:question:) for questions
• 🤔 (:thinking:) or 💭 (:thought_balloon:) for more open inquiry that's not quite a confirmed issue and could potentially benefit from discussion
• 🎨 (:art:) for suggestions / improvements
• ❌ (:x:) or ⚠️ (:warning:) for more significant problems or concerns needing attention
• 🌱 (:seedling:) or ♻️ (:recycle:) for future improvements or indications of technical debt
• ⛏ (:pick:) for minor or nitpick changes

[github-actions]

Checkmarx One – Scan Summary & Details – 3a146bc5-326b-4a01-ba09-962f1d0b2dcc

Great job, no security vulnerabilities found in this Pull Request

[codecov]

Codecov Report

Attention: Patch coverage is 25.00000% with 3 lines in your changes missing coverage. Please review.

Project coverage is 36.48%. Comparing base (23f4af6) to head (1b407ed).
Report is 19 commits behind head on main.

✅ All tests successful. No failed tests found.

Files with missing lines	Patch %	Lines
...pps/browser/src/platform/listeners/update-badge.ts	0.00%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main   #14709      +/-   ##
==========================================
- Coverage   36.49%   36.48%   -0.01%     
==========================================
  Files        3207     3207              
  Lines       93242    93241       -1     
  Branches    16833    16833              
==========================================
- Hits        34027    34020       -7     
- Misses      56742    56748       +6     
  Partials     2473     2473              

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[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud