[Reporting] Fix Headless Browser Console & Process logging

[tsullivan]

Summary

Closes #43295

Checklist

Use strikethroughs to remove checklist items you don't feel are applicable to this PR.

• This was checked for cross-browser compatibility, including a check against IE11
• Any text added follows EUI's writing guidelines, uses sentence case text and includes i18n support
• Documentation was added for features that require explanation or tutorials
• Unit or functional tests were updated or added to match the most common scenarios
• This was checked for keyboard-only and screenreader accessibility

For maintainers

• This was checked for breaking API changes and was labeled appropriately
• This includes a feature addition or change that requires a release note and was labeled appropriately

[elasticmachine]

💔 Build Failed

• continuous-integration/kibana-ci/pull-request

[elasticmachine]

💚 Build Succeeded

• continuous-integration/kibana-ci/pull-request

[elasticmachine]

💔 Build Failed

• continuous-integration/kibana-ci/pull-request

[tsullivan]

replace with #44056