Hotkey for automator service without giving permissions to all applications
I had the same issue. What worked for me: 

1: Saving the automator workflow as an application (File > Convert To > Application. Save > File Format = Application). 

2: Then creating a separate Automator Quick Action with the sole purpose of launching the application created in 1 (File > New > Quick Action. Save normally as a workflow.). 

The application containing the target workflow (1) will then need to be given Accessibility permissions via the usual (Security and Privacy > Privacy > Accessibility) and the key shortcut should be bound to the workflow (2) as you described. 

This successfully bypasses the security settings of the currently active application by running the workflow in a self-contained app (with its own Accessibility preferences) negating the need for multiple permissions.