There is also this old blog post by Yegge [1] which mentions `AXUIElementPostKeyboardEvent` but there were plenty of bugs with that, and I haven't seen anyone else build on it. I guess the modern equivalent is `CGEventPostToPSN`/`CGEventPostToPid`. I guess it's a good candidate though, perhaps the Sky team they acquired knows the right private APIs to use to get this working.
Edit: The thread at [2] also has some interesting tidbits, such as Automator.app having "Watch Me Do" which can also do this, and a CLI tool that claims to use the CGEventPostToPid API [3]. Maybe there's more ways to do it than I realized.
[1] https://steve-yegge.blogspot.com/2008/04/settling-osx-focus-... [2] https://www.macscripter.net/t/keystroke-to-background-app-as... [3] https://github.com/socsieng/sendkeys