📝 Docs: Overhaul website generation

[JoshuaKGoldberg]

Documentation Request Checklist

• I have read and agree to Mocha's Code of Conduct and Contributing Guidelines
• I have searched for related issues and issues with the faq label, but none matched my issue.
• I want to provide a PR to resolve this

Overview

Back in #5027, we stated:

We don't intend to overhaul any currently-useful resources such as https://mochajs.org/ or https://github.com/mochajs/mocha/wiki. Though, we will happily accept issue reports if those resources need improvement.

We still don't want to overhaul the contents of the Mocha website. However, the tooling behind the website is ... not great to work with. It's built with the very old https://github.com/mochajs/mocha-docdash which comes with a ton of old dependencies that are hard to update (#5071).

We'd like to redo the website's tooling to be something more modern. I'm personally partial to Astro Starlight. I've had great experiences with Docusaurus too.

Additional Info

This is roughly blocking #5055. If we enable Renovate, we'll get a bunch of update PRs for the same packages touched by #5071. Irksome.