Add test MQTT subscription is completed when birth message is sent

[jbouwh]

Proposed change

Add test MQTT subscription is completed when birth message is sent

As an addition to: #116471

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

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• This PR fixes or closes issue: fixes #
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[home-assistant]

Hey there @emontnemery, @bdraco, mind taking a look at this pull request as it has been labeled with an integration (mqtt) you are listed as a code owner for? Thanks!

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[bdraco]

asyncio.timeout is generally preferred over asyncio.wait_for since its more efficient

[bdraco]

This doesn't generally matter anymore though since asyncio.wait_for is effectively a wrapper around asyncio.timeout in py3.12+

[bdraco]

Tagging this to make it easier on whoever is doing the release by avoiding future conflicts in case we have to backport any more mqtt fixes to 2024.5.x

[bdraco]

Thanks @jbouwh