Not all anarchists are teenage rioters who just want to make trouble and listen to punk music. Hell, many (most?) are not even interested in an active (let alone violent...) overthrow, and are no more politically active than any other person of more mainstream political ideologies.
What you are proposing is that the government spy on anyone who falls within a broad category of political classification. I think the issue with that should be self-evident.
"Our aim is not to overthrow the state, but to ignore it. Anyone who wants to continue to support the state and obey its laws is free to do so, so long as they leave us alone. Our goal is to build the kind of society we want, and prevent the state from overthrowing us while we’re doing it. The last person out of the state can turn off the lights." - Kevin Carson
To some anarchism is the idea of a completely voluntary society. A violent overthrow of the state is against this idea.
What you are proposing is that the government spy on anyone who falls within a broad category of political classification. I think the issue with that should be self-evident.