They were the embodiment and a measurement of the imperium which gave authority over government and military command.
Imperium was by the nature of Roman law extremely limited and specific. What you're saying has no relevance to the authority of a magistrate to make detentions of citizens on the grounds of criminal or civil wrongdoing.
Getting back to it, this meme appears to require a very specific and literal use of ‘cop’ to work.
"Getting back to it"
Your entire point is that lictors are, in your view, cops, so there's no 'back to it' involved.
If you feel so strongly about the word usage, please define 'cop' for me in such a way that would not imply that anarchist polities also have 'cops'.