Fix MQTT discovery cooldown too short with large setup

[jbouwh]

Proposed change

Change the MQTT discovery cooldown from 2 to 5 sec. This avoids the birth message to be published too soon.
When we have processed the subscriptions we ensure they are executed immediately after discovery cooldown.

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

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue: MQTT does not start up properly (sometimes) #115958
• Link to documentation pull request:

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• The code change is tested and works locally.
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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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[jbouwh]

Thnx!