fix(profile): adjust profile handling at the end of the game

[Joltras]

Proposed changes

Fixes game end handling to ensure final statistics and winner title announcements are correctly delivered to all online players. Ensures player profiles and active tasks are properly cleaned up at the end of a match to prevent memory leaks.
Corrects the winner calculation logic and includes integration tests to verify game finish behavior.

Types of changes

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• Bugfix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)
• Documentation Update (if none of the other choices apply)

Checklist

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• I have read the CONTRIBUTING.md
• I have added tests that prove my fix is effective or that my feature works
• I have added necessary documentation (if appropriate)

[github-actions]

Test results

 84 files   84 suites   1m 2s ⏱️
 60 tests  60 ✅ 0 💤 0 ❌
180 runs  179 ✅ 1 💤 0 ❌

Results for commit 4362f84.