Wordpress site health check critical issues caused by CiviCRM plugin (REST API, cURL, loopback)
It the CiviCRM plugin is enabled in WordPress, the site health check (Dashboard > Tools > Site Health) immediately reports two critical issues:
The REST API request failed due to an error. ERROR:  cURL error 28: Operation timed out after 10002 milliseconds with 0 bytes received
The loopback request to your site failed, this means features relying on them are not currently working as expected.
The only immediate practical consequence I have found is that the theme editor (Dashboard > Appearance > Theme Editor) no longer works - if you try to edit a file (e.g. header.php) and save it by clicking "Update File", nothing happens except a spinning progress indicator. Since there are other (better) ways of editing theme files this is not a high priority issue, but it's not good to be seen causing "critical" errors.
This issue (and a possible solution) are described in detail here and in linked answers - it's caused by using PHP sessions. https://wordpress.org/support/topic/the-loopback-request-to-your-site-failed-4/page/2/