[WIP] docs: switched API docs to typedoc and starlight-typedoc

[JoshuaKGoldberg]

PR Checklist

• [x] Addresses an existing open issue: fixes #5309
• [x] That issue was marked as status: accepting prs
• [x] Steps in CONTRIBUTING.md were taken

Overview

Very much a draft WIP. Both the current docdash/jsdoc docs and the new TypeDoc generators generate some wacky code. My hope is we can clean up the current docs to be better for both.