🛠 Tooling: dist/ web game directories should be fully portable

[JoshuaKGoldberg]

Tooling Report Checklist

• I have tried restarting my IDE and the issue persists.
• I have pulled the latest main branch of the repository.
• I have searched for related issues and found none that matched my issue.

Overview

It should be the case that you can open a game's dist/index.html anywhere:

git clone https://github.com/JoshuaKGoldberg/ChooseYourFramework
cd ChooseYourFramework
yarn hydrate
yarn dist
open dist/index.html

...but, alas, that is not yet the case. Let's fix up those paths. Any change to href/src/etc. paths from https://github.com/JoshuaKGoldberg/ChooseYourFramework/blob/0a18b19632b5f321e990748ce8ddb9b5588e005f/fix-index-for-vercel.js#L18 should be incorporated into shenanigans-manager.

Additional Info

Consider this split out of #311.