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I'd imagine other similar type third-party applications that allowed assigning keyboard shortcuts and running a script would bypass the issue too, but have only tested the aforementioned. The first, my preferred method for running AppleScript scripts with a keyboard shortcut, is to use a third-party application named FastScripts, as it would not need to have every application that's frontmost, that hasn't yet been granted privileges, to be granted accessibility privileges to run the AppleScript code shown in your question. To workaround the accessibility privileges issue, here are three methods to achieve the goal that come to mind.

Every application that is frontmost when the keyboard shortcut is pressed on a Service/Quick Action using UI Scripting in the Run AppleScript action will need to have accessibility privileges granted for it (as you've already found this out and hence the question).The keyboard shortcut assigned to the Service/Quick Action needs to not conflict with a default keyboard shortcut for whichever application is frontmost at the time it's pressed, otherwise there may be unwanted behavior.

There are issues at play here with using global keyboard shortcuts with Automator workflows saved as a Service/Quick Action.

Automator mac script leave imessage conversation