[speleding]

You can use dig:

  dig +short -4 myip.opendns.com @resolver1.opendns.com

(still an external service, but not reliant on a website)

[lxgr]

That's just another protocol in the end (DNS vs. HTTP), or am I missing something?

DNS also runs a higher risk of being cached or mangled somewhere along the way by middleboxes. HTTPS avoids that problem.