Fix mqtt reloading and rework tests

[jbouwh]

Proposed change

Fix reloading issues for MQTT button and select platform, and add tests for reloading on all supported platforms.

Reloading support was initiated for MQTT with #39531. Tests were only added for the light platform.

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)
• 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 Server Controls -> YAML config reloading -> Manually configured MQTT entities does not work for new Button entities #62155
• This PR is related to issue:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
• The code has been formatted using Black (black --fast homeassistant tests)
• Tests have been added to verify that the new code works.

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated for www.home-assistant.io

If the code communicates with devices, web services, or third-party tools:

• The manifest file has all fields filled out correctly.
  Updated and included derived files by running: python3 -m script.hassfest.
• New or updated dependencies have been added to requirements_all.txt.
  Updated by running python3 -m script.gen_requirements_all.
• For the updated dependencies - a link to the changelog, or at minimum a diff between library versions is added to the PR description.
• Untested files have been added to .coveragerc.

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[probot-home-assistant]

Hey there @emontnemery, 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!
_{^{(message by CodeOwnersMention)}}

[emontnemery]

Looks good, can be merged when the merge conflict is solved 👍