Remove connection counts and limits from API health checks (backport #13888)

[mergify]

Proposed Changes

This is a minor change to remove the connection limit and current active count from health checks which returned them (introduced in #13879).

Returning the connection limit and active count are not really necessary for these checks. Instead of returning them in the response to the health check we log a warning when the connection limit is exceeded.

Connects #13782

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• Bug fix (non-breaking change which fixes issue #NNNN)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause an observable behavior change in existing systems)
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• Cosmetic change (whitespace, formatting, etc)
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This is an automatic backport of pull request #13888 done by [Mergify](https://mergify.com).