Whitelist your User Agent and it should fix 🙂
This one took me FOREVER to track down. But the problem ending up being in the WP core wp-includes/general-template.php function user_can_richedit().
the function looks at what browser is being used by the request user agent headers. Cloudfront was sending “Amazon Cloudfront” as the user agent which was blocking tabs from being echoed out for the classic editor.