Do you absolutely have to go through your Domain Name registrar to get a public IP registered as the authorative name server for your domain??

yes you have to go through your Domain Name registrar

So-called 'child' nameservers (ns1/ns2/ns3/etc.yourdomain.tld) are kept and managed by the registry, so *theoretically* you can go via registry; however I strongly doubt they will speak with you.

So, short answer is: yes, you must go via your registrar to create nameserver(s) for your domain.