Add CI to check the nix development environment builds successfully

This issue has been migrated from [#15941](https://github.com/matrix-org/synapse/pull/15941).

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Ideally this would be a nightly job, so that if it fails on PRs this doesn't block anything.

Note to self: GitHub Actions steps each appear to start a new environment. Write a script that starts with `nix develop --impure` and takes as an argument what test suite to run.

Start by reviewing the repository's GitHub Actions configuration and the Nix development environment, then check how `nix develop --impure` can invoke the requested test suite. Done means a nightly CI job verifies the development environment builds successfully without blocking pull requests.

Written by the indexing model from the issue text.