🧹 Cleanup: Rewrite InputWritr to use generics and have less convoluted settings

[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

InputWritr is one of the oldest modules in this package and the design shows it. Two main issues:

• The system of events/triggers/types and their aliases is convoluted and hard to understand
• Many of the types use strings instead of type parameterized unions, and/or as any internally

I'd like to revisit the design from scratch to make full use of newer TypeScript features. And then also be comprehensible.

Additional Info

No response