Ah, well, yes, violence will get you what you want in the short term, but unless you exterminate your enemies, they'll still be a liability: violence doesn't convince anyone, it just makes them more tractable.

Let's frame it as comparing two options against each other (not one option against some unspecified ideal outcome). Based on a quantum event, in one branch of the multiverse you use violence, in another you refrain from violence. Is your long-term utility in the first branch necessarily lower than in the second branch?

Your enemies are not a homogenous group. Some of them are very convinced and eager to act, but others just like to join the existing group and be an anonymous part of the mob. Destroy the former, and you don't have to worry so much about the latter.

A short-term advantage can sometimes be used as a leverage for long-term advantage. If you happen to rule a country, you can change education and media to include your propaganda. This increases the number of your followers in long term. (Back to typical LW topics: If you are an AI capable of recursive self-improvement, paralysing your opponents for a week may be all you need to conquer the universe.)

Depending on specific circumstances... for example, your enemies may be popular because they have an aura of invincibility. Beat them once and you have ruined the aura, which may remove many of their followers.

(Disclaimer: I don't suggest using violence, I just say that sometimes is may be the better choice, so the "violence doesn't achieve anything" is untrue.)