Serverless and static are different. Static sites will likely be limited to maintaining state in the browser. Serverless is terribly named but still have a short lived server with access to request query parameters, cookies, headers, and a database depending on your setup
You can embed a "backend" as a service worker and use htmx to interact with it.
A proof of concept here: https://github.com/richardanaya/wasm-service
This is cool!
A possible extension to HTMX would be to allow this kind of offloading to pure JS functions instead of requiring hacky intercepts.
You would still have a clear separation of responsibilities between frontend rendering (by the browser only) and application logic (which only generates HTML as output).
I've done the same thing with vanilla JS in a service worker. I did it with my HTMF library -- similar to HTMX -- but not production ready and based on forms rather than being able to put your attributes everywhere. I'll have to create a simple to do app with HTMX that shows this same functionality. The Rust one is cool as you can do it with 150kB library. But I can do the same thing with 10kB with just simple JS.
In a serverless read-only app, all business logic and state is maintained on the browser.