Port 444 is not typically a port that serves websites. It is conceivable, for example, that your network security team might be willing to add a firewall rule allowing you to listen on a privileged port other than 80/443 for some service. They probably won't suspect that this would also give you the means to acquire a certificate for their domain. For use-cases where existing web server configuration cannot be modified or port 80/443 cannot be used, there's still DNS-01.
As an aside, the CA/B Forum is currently working on updating the rules for domain validation (still a WIP). They have defined a list of ports that are acceptable for this purpose:
Authorized Port: One of the following ports: 80 (http), 443 (http), 115 (sftp), 25 (smtp), 22 (ssh).
ACME/Let's Encrypt would have to stick to those once those new rules pass the ballot vote. I haven't seen any discussion about adding the last three to ACME on the WG mailing list, but I suppose it would be possible/could be brought up if anyone feels like it's a good idea.