Focus search input on integrations dashboard when using search shotcut

[jpbede]

Proposed change

Also focus the search input on the integrations dashboard page when using the search shortcut as we do on the data table pages.

Type of change

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

Example configuration

Additional information

• This PR fixes or closes issue: fixes Ctrl-F does not set focus on search box in Integration page (while doing it in the other three tabs on same page) #23631
• This PR is related to issue or discussion:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
• There is no commented out code in this PR.
• 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

[silamon]

CI is failing, but unrelated.