🚀 Feature: Use Prettier when outputting files instead of JSON.stringify

### Feature Request Checklist

- [x] I have pulled the latest `main` branch of the repository.
- [x] I have [searched for related issues](https://github.com/JoshuaKGoldberg/create-typescript-app/issues?q=is%3Aissue) and found none that matched my issue.

### Overview

I'm finding it annoyingly difficult to read snapshots of generated `package.json` files:

https://github.com/JoshuaKGoldberg/create-typescript-app/blob/2396960cb7a23acc6a9952040f203a8543cc2266/src/blocks/blockPackageJson.test.ts#L53

It'd be nice to use [`@prettier/sync`](https://www.npmjs.com/package/@prettier/sync) when producing file creations. That way their test snapshots look human-readable.

They're formatted at runtime with Prettier anyway if the user doesn't disable `blockPrettier`. I've never seen a user disable that block. But it's theoretically possible.

### Additional Info

Previously filed as https://github.com/JoshuaKGoldberg/create-typescript-app/issues/2044. The problem with `JSON.stringify` is:

1. It adds `\n`s in many more cases than Prettier
2. Prettier will pick up on those cases and preserve them in its own formatting
3. As a result, some output files have unnecessary `\n`s that make them take more lines than needed

Start with src/blocks/blockPackageJson.test.ts, especially the snapshot around the linked lines, and trace how generated file contents currently use JSON.stringify. Review the @prettier/sync and blockPrettier context before running the relevant block tests. Done means generated package.json snapshots are human-readable without unnecessary newlines while existing formatting behavior remains covered.

Written by the indexing model from the issue text.