Update eslint configuration to use new flat config instead of rushstack

[rtibbles]

The issue that motivated rushstack/eslint has some recent updates from two weeks ago:

Hi folks! Just an update that the new ESLint configuration has now been enabled as an experiment. This new config system eliminates the special treatment of shared configs so that you can include plugins as direct dependencies. Read about it here:

https://eslint.org/blog/2022/08/new-config-system-part-1/
https://eslint.org/blog/2022/08/new-config-system-part-2/
https://eslint.org/blog/2022/08/new-config-system-part-3/

This is the long-term solution to the problem in this issue. We know it took a while, but this was a really complex problem that we needed to think through.

Because we now have this solution implemented, I'm closing this issue. Please try out the new config system and leave your feedback in the discussions.

Originally posted by @MisRob in #9698 (comment)