Got it. The generator logic is in Rust yes. I could have written it in JavaScript as well but it was just easier for me to do that part in Rust. Should have mentioned that.
Of course you could host a static HTML with JS on Github pages, but Github pages also runs on a server. So to answer the original question, yes you do need some server for it unless you make it a command line app.
/me glances at serverless in the title. Man, I would not want to have to technically design anything with you. You can't just say a thing is something and then say "oh well it's not actually but it could've been if I had made it entirely differently." ...
You can serve a static HTML page from anywhere. GitHub pages, S3, maybe even Dropbox. Running a Rust binary is an order of magnitude more complex than that.