Fully migrate to the new eslint configuration.

[brian-mcallister-lab49]

Probably not something we want to do right away, but I've taken the first steps towards this. There's more documentation on this page: https://eslint.org/docs/latest/user-guide/configuring/configuration-files-new. Here's a screenshot as of this writing:

Right now, I do have the ESLINT_USE_FLAT_CONFIG env var set to false so that we can maintain compatibility with other tools. There's additional setup in the repo to keep things working, but should be cleaned up.

For example, we're needing to set these VS Code preferences to keep things functional: microsoft/vscode-eslint#1518 (comment)

I don't know what would be required or available in other IDEs.