Fixes some issues with the Checking icon I added last week.

- Use default text color for "Checking..." text and icon
- Always show text and icon while checks are running

I think we should probably remove the `✅` while checks are running as it
incorrectly indicates nothing is wrong when we don't yet know that – but
that is a larger change so I'll keep this fix contained.

Release Notes:

- Fixes "Checking" diagnostics icon only showing until the first
diagnostics item is found.