Yes - I think the key to effective usage really is to keep the number of shortcuts to a minimum for the things you need to be most accessible if you're physically pushing the button. If you end up with hundreds of shortcuts sprawled in a UI, you'll likely spend more time searching for the button than is tolerable. Of course, if you're going to use your voice, then you can simply keep all the shortcuts hidden away in the shortcuts app and talk to Siri.
Unless both phones are signed in using the same iCloud account, or you've got some kind of Family Sharing setting in play, I'm not sure either: usually you'd need to manually share them. Are they both on separate iCloud accounts?
Yes, my view is that your issue isn't with Cortex - this inbound route works and is proven by your iOS Shortcuts working, so the issue has to be in the forming of the request in your Applescript. Whack some tracing between the Applescript and Cortex to check where it's going wrong.