To be “public” you’ll need a web domain to which users can be linked, where you should also have policies and disclosures about data retention if you use actions.
Having a pre-paid API account lets you use the OpenAI API services that are billed by amount of data use. It doesn’t impart anything onto ChatGPT.
Domain verification procedures, for those somewhat versed in web setup, and of course having their own domain name under their control: