I also have a feeling that maybe there is something wrong with the default configuration? Maybe there is a limit set on how many requests a single user can make in a second or so
If you have any sort of rate limiting or security software, you should make sure you allowlist Cloudflare’s IP ranges: IP Ranges as well as restore visitor IPs.
Surely you should have access and be able to review the error logs in apache/nginx as well as PHP and cPanel. Check those and see if any of them show why the 503 error is happening sometimes.
The php log would probably be it, but if you’re using Apache IIRC the PHP error log is a part of the Apache error log.
Is the cpanel server on a separate subdomain/domain? If so, it may be worth it to temporarily disable the proxy (click the to turn to ) on that single subdomain to see if CF is the cause.