Does anyone know of a "general" static site generator, where there are no such preconceived notions?
I only want to generate a number of HTML pages, based on an arbitrary number of templates.
Static site generators take input, run it through templates, and produce html output. This seems to be, literally, what you want to do. Maybe try and implement what you want and you'll see they can do it.
I use Hugo to generate some static pages that sit inside a dynamic website. It's just Go templating, which is not too hard to learn. You can create individual pages that pull in multiple templates if you want. You can switch off RSS, change naming from posts to bananas, etc. Very few preconceived regulations.
Most of the static generators are used for blogs, that's probably why most examples they give are around blog posts. It doesn't mean the name 'post' actually means anything special.
The reason I prefer Org for static site generation is not because I already use Emacs. I actually started using Emacs about 20 years ago specifically to use Org mode.
For example, if you're working in the React world, Next.js can actually output static HTML pages that work fine without JS... just use the pages router and a static export (https://nextjs.org/docs/pages/guides/static-exports). That still lets you use all the power of JS and expressiveness of React components, minus the interactivity, of course (if you don't want JS). But you could still pass in components and such. It's a bit like writing serverside includes in the PHP or Perl days. The benefit of using Next is its incredible popularity; probably whatever question you have, someone else has already asked and ten people have answered it. The downsides are its complexity and its frequent changes; answers from just a year or two ago are probably irrelevant to the current version, and there is a steep learning curve at first. But in SSG mode with the pages router, it's pretty straightforward, and the filesystem-based routing makes it very clear what the final directory structure would be.
For Markdown there's https://docusaurus.io/ and https://github.com/rust-lang/mdBook
Eleventy is another option for various templating languages: https://www.11ty.dev/
See also these lists: