Hi there,
Your setup is correct but you forgot to delete a few old MX records. This is what I saw on your server now:
20 eforward5.registrar-servers.com.
15 eforward4.registrar-servers.com.
10 eforward1.registrar-servers.com.
10 eforward2.registrar-servers.com.
10 eforward3.registrar-servers.com.
10 mx1.privateemail.com.
10 mx2.privateemail.com.
you should delete all other MX records in Cloudflare except mx1.privateemail.com and mx2.privatemail.com
Why? Because you original buy the domain in namecheap, they have built-in email forwarding which set those records:
20 eforward5.registrar-servers.com.
15 eforward4.registrar-servers.com.
10 eforward1.registrar-servers.com.
10 eforward2.registrar-servers.com.
10 eforward3.registrar-servers.com.
Then once you add private email, you didn’t delete them. so now it has 2 systems that are receiving your emails. When you are testing from your phon, you get lucky and it goes to mx1.privatemail.com or mx2.privatemail.com and you can see email. But when you try from other, they are routed to eforward3.registrar-servers.com etc, and those server rejected the emails because you are no longer use namecheap DNS.