shenanigans-manager: Add real-time TypeScript transpiles for tests

[JoshuaKGoldberg]

Right now, the testing infrastructure is annoyingly two-step:

1. Source files and tests written in TypeScript must be transpiled to JavaScript
2. Tests can then be run in-browser or on the CLI via mocha-headless-chrome

Ideally (1.) would be completely removed so that test files don't need to be transpiled before running. That'd make testing a bit easier to do:

• No more yarn compile -w in order to prepare test files to run
• CI builds could run tests in parallel with compile, instead of in series in the same task

There are three likely possible ways to improve this situation, in order of my preference:

• Targeting more than JSDOM jestjs/jest#848: Jest could support running in a browser (Jest is my personal favorite test runner)
• In theory, Mocha might be able to directly import and transform test files the way Jest does - though it's more complicated because of mocha-headless-chrome (ts-node is not the answer)
• shenanigans-manager could switch to another test runner such as Karma

I would accept any of the above three things as a resolution to this issue - though switching to something like Karma is the most likely. It's acceptable to me that rewriting test assertions to a different library such as Jasmine would be required.