I use and recommend Lume [1] for static site generation, and I know it has a couple of search plugins, but I haven't used them. As others have mentioned, your search without JS requirement will be hard to implement, unless you use an external provider for the indexing and search.
"Search without JS" and "static site generation" doesn't make much sense. Without client-side search, you'll need a search product or back-end search functionality? (both of which are not "static")
Or maybe another option would be to outsource the search part to "site:yourdomain" on google or something similar.
[1] https://lume.land
The whole site can be configured in the astro.config.js file and the rest is markdown.
If you want an example you can look at sst.dev and their github repo at https://github.com/anomalyco/sst/tree/dev/www
It's been super easy and I host it free on CF pages:
[1]https://hugo-book-demo.netlify.app/docs/example/ [2]https://docs.formbeep.com
It generates static pages using only HTML and CSS, which makes it blazing fast. It can also handle jQuery and JavaScript very smoothly.