I've been annoyed with super slow domain availability checkers, and their suggestions.
So I built something that uses DNS resolve requests to check for availability and sends the typed query in real time to gpt-3.5-turbo for instant suggestions: https://einstein.domains/
It is possible to modify the default prompt to tell it to be more specific and guide the process.
I made the whole thing around performance. Some requests take less than 5 ms. Of course, DNS Resolve requests are only 95% accurate. Sometimes a domain is reserved by the registrar (thus unavailable) but has no DNS, and thus produces a false positive. But the gain in performance is so high, that it's an acceptable error-rate for my use case. It's always possible to click "Buy" and see if the domain is actually available.
Hope this is useful to you. I built it for myself, but please enjoy. Whether you want to use it as the fastest way to check domain availability, or do some brainstorming.