We use our own system for Custom Colors. This system gets the main style CSS file content and creates new generated styles file under ../wp-content/uploads/ folder. If the Custom Colors system is not working on your site, you need to check below options:
Disable all caching and optimizing plugins in your site (CDN, CloudFlare, and Akamaisystems also).
Check if your hosting supports PHP FILE extension.
At the end, check if WP_Filesystem is working correctly. You can check it using below code in your functions.php. Ifprint_r() doesn't show any results,it means WP_Filesystem is not working on your site. In this case, you can contact your hosting providers with this issue.