📝 Docs: Switch the new website to be the primary one

[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

Eventually, we're going to want to switch the new docs website to be the default / primary one seen on https://mochajs.org. Very exciting. But, two blockers:

• It'll need to go through some user testing first: at least after 📝 Docs: Add a banner/notice to the current site linking to the new site #5304
• There are a collection of bugs / missing features / proofreading issues that we'll want to resolve first

...so while this issue will happen eventually, it's blocked for now.

Additional Info

#5246 has a description of the rough plan I was thinking of while working on the PR. https://github.com/orgs/mochajs/projects/8/views/1 is a more full milestone with that plan fleshed out.