I just saw that OpenAI released an official Chat GPT app for iOS so I downloaded it tonight and gave it a spin. My first impression is very good! It's just about as accessible, simple and logically organized as you could hope for. I found it much easier to use than their web interface which I find has some major accessibility pitfalls.
Of particular usefulness is the ability to search your chat history which I don't believe the website offers. It also uses haptic feedback to let you know that a response is still being generated.
Anyway, if you're a Chat GPT user and you've also found the site kind of a pain to use with a screen reader, the iOS app seems well worth giving a try. Cheers.