fix: override # private unions in Horror Factory

[JoshuaKGoldberg]

PR Checklist

• [x] Addresses an existing open issue: fixes #276
  • There might be more work to do for tests (I asked in https://github.com/LearningTypeScript/projects/issues/276#issuecomment-1720016386), but we can tackle that separately
• [x] That issue was marked as accepting prs
• [x] Steps in CONTRIBUTING.md were taken

Overview

Uses an as assertion to intersect the type of the index file on top of the solution file. In other words, this tells TypeScript that the type of the solution file should also include everything from index, making the classes assignable to each other despite having different # privates.