Take a look at mkdocs-material, it's not just a theme, it adds some local-side search as well. Still with fully static pages.
Configuration in https://gitlab.com/dnsmichi/opsindev.news/-/blob/main/mkdocs...
Material for MkDocs also has an insiders build, accessible through sponsorship. https://squidfunk.github.io/mkdocs-material/insiders/ These features add more value to MkDocs -- I initially joined to get GDPR-compliant cookie banners and stayed to support a great project.