See also:
https://developer.apple.com/documentation/sirikit#2979425
https://support.apple.com/en-gb/guide/shortcuts/welcome/ios

Yes a "proper way" to do this would really be useful.

(The workflow:// method described below won't work for me).

This can be done using the workflow:// scheme. Tested it and works. Assign your flic to open safari and add url eg. workflow://run-workflow?name=NameOfShortcut&input=input