Mail elsewhere, website with us.
WHAT TO CHANGE
MX records pointing at the provider's servers, with their priorities, exactly as specified.
SPF including the provider.
DKIM records the provider supplies.
Any additional records they require.
THE CPANEL SETTING
Email routing must be set to remote for that domain.
If it remains local, mail sent from the server to an address on your domain is delivered to a local mailbox instead of being sent to the external provider.
The sender sees no error. The message simply never arrives.
That is the failure people cannot explain, and it is the first thing to check.
WHAT ELSE TO CONSIDER
Existing local mailboxes. If you no longer use them, remove them to avoid confusion. Your website's outgoing mail, which may still send through the server. Configure SMTP through the external provider if you want everything consistent.
TESTING
Send to the domain from outside and confirm arrival Send from the website's contact form and confirm arrival Send from a mailbox and check authentication