Modules documentation is confusing/absent/outdated/broken

How do I make a (local) version of element-web, with extra modules enabled? I tried to find some documentation, but ended up in a twisty maze of confusion.

* [ ] https://github.com/element-hq/element-web/blob/develop/packages/module-api/README.md describes the API, but not how to build a module that uses said API.
* [ ] https://github.com/element-hq/element-modules-private (private repo) contains some modules, but doesn't tell me how to use them in EW.
* [ ] The "about" box in https://github.com/element-hq/element-modules-private links to https://github.com/element-hq/element-modules, but I think that repo is outdated?
* [ ] If https://github.com/element-hq/element-modules is actually outdated, could the README say so?
* [ ] https://github.com/element-hq/element-modules contains some modules (well, just `opendesk`) but doesn't tell me how to use that in EW.
* [ ] https://github.com/element-hq/element-web/tree/develop/modules also contains some modules, but doesn't tell me how to use them in EW.
* [ ] I found https://github.com/element-hq/element-web/blob/develop/apps/web/build_config.sample.yaml, but attempting to use it seemed to break the entire build.
* [ ] https://github.com/element-hq/element-web/blob/develop/apps/web/build_config.sample.yaml mentions a file called `./docs/modules.md` which no longer seems to exist. (https://github.com/element-hq/element-web/blob/develop/docs/deprecated-modules.md maybe?)
* [ ] https://github.com/element-hq/element-web/blob/develop/docs/deprecated-modules.md could do with explaining a bit more about what the difference between the old an new system is. (Build-time vs run-time modules, maybe?)
* [ ] https://handbook.element.io/books/element-web-team/page/element-modules (internal documentation page) looks reasonably up-to-date at least, but contains a bunch of dead links and still doesn't actually tell me how to use the modules.

It would be lovely if we could have a doc in the repo that gives:
* a basic overview of the new module system
* how it differs from the old system(s)
* how to use modules

... which we could then link to from the various places above.

Start by reading packages/module-api/README.md, modules/, apps/web/build_config.sample.yaml, and docs/deprecated-modules.md, then compare the linked module repositories and handbook page. Done means a repository document explains the new module system, its differences from the old system, and how to use modules in Element Web, while identifying outdated or broken references.

Written by the indexing model from the issue text.