First, arguments from your opponents stated motivation are obviously flawed. He is using an inappropriate notion of cause, and therefore of solution:
Follow the argument down to its logical closure: It is likely given the number of minor infraction for which, via this argument, would get you wounded or killed, that basically no one would want to live in such a world. The level of stress and fear would simply be too high. Following the proposed pattern, if no want wanted things this way, we would all just kill ourselves. Problem solved? No. This points up the fact that eliminating a problem is not solving it.
Where does the distinction arise? Fairly early: If I decide to solve a mechanical problem in my car by disposing of the car, I am not solving the problem. The problem is that I do not have a functioning car, and after disposing of the car, I still do not have a functioning car. We need a definition of 'solved' that makes sense, and that involves discerning the functional failure of the system and addressing it.
But thwarting one argument against something does very little toward showing it is actually false.
There might still be some other way in which violence might be applied to every problem. To really undermine the idea that violence solves every problem, you need an important example where violence never really helps. (If it helped at all, it might still be the lynchpin of the application, the thing that tips the balance, breaks the final impasse, and therefore really solves the problem.)
Getting correct information from other people is such an example. Nor is it a trivial example. Much of the functional failure in our society is about ineffective communication. And much of that failure to communicate is based in fear.
For instance, leaders arrange to control the knowledge of their acts so that they will not be held accountable, and the decisions they make as a result create opportunities for corruption and political inefficiency. Producers make sure that others do not know how to truly evaluate the quality of their products so that consequences of lower quality seem accidental and are not traced back to them. This prevents honest competition and makes markets inefficient, as well. So failure to communicate truthfully is a major problem for any political system.
This fear works in a way that is not amenable to solution by creating additional fear. Since most violence short of murder does raise the level of fear, you are caught in a place where violence cannot help you. (And, obviously, killing someone permanently prevents getting any further information out of them.)
To see how adding fear does not help, look at torture. We know from studies of torture that additional fear or pain motivates one to give information, but it primarily motivates lying. If someone will give you truthful information under duress, they would generally have given it to you without the use of violence.
This is disputed at great length, and lied about in Congress and in fiction, but the data are clear. No matter how you abuse or threaten someone, with direct pain, psychological force, threats to loved ones, etc., the primary effect is to garner false information, mostly false confessions or false implications of the guilt of innocent people.
Other uses of violence to get reliable information suffer the same intrinsic failure -- it is far too easy to lie; lying preserves the speaker's worth just as much as, and often more than, telling the truth; and lying makes one feel like one has some control, some ability to thwart the person harming you and get a minimal revenge. So, knowing that adding fear primarily creates lies, instead of more truthful communication, in what way can violence solve the huge number of problems traceable to this single cause?
Since we cannot get better information from others by violence, if violence is to solve our information-related problems, it would have to be by preventing there being information we must acquire from others. We can only have privileged access to all information through some form of totalitarian surveillance.
But again, the data are clear. We have studied police states and surveillance in cults. To the degree that actual violence is a part of surveillance, it creates fear of the consequences of misunderstandings. Even if you are innocent, you can look guilty and get punished. People act on this fear, causing subterfuge, sabotage and rampant basic dishonesty to undercut the effectiveness of the surveillance.
The logical reaction is to increase the consequences of lying, but enforcing those consequences would require knowing who is lying, which involves getting information that is already being withheld effectively. So there is no actual point of application that will make it a better risk for everyone to tell the truth.
So, the problem remains unsolved, and it is hard to imagine better applications of violence that have not been ruled out by what we already know about how people naturally react to being controlled with violence. (We know a lot. We have been doing it for a long time.)
You can escape this by defining down violence to more subtle forms. For example cults can achieve very complete control of truth, and surveillance through authority, guilt, and fanaticism, as long as the enforcement is not seen as violence. Once it is, members naturally collude in resisting it.
But that is just cheating. Social coercion and more subtle means of control simply are not violence. So even in the most bloody-minded, psychopathic picture of the world, violence fails us in addressing one important range of problems.
(Obviously, I don't personally accept this psychopathic, war-of-all-upon-all worldview. But it is part of the general argument for violence, so it is a better frame for an argument toward contradiction.)
But the opposite extreme argument, that violence is unnecessary and can be done away with, which is what you presumably want help defending, is also flawed:
If you don't have organized violence, given that agreement is never total and destruction is always an option you cannot take away from people, you will have disorganized violence. And it may be preferable to go that direction, but then that is not a society where violence is unnecessary, it is one where it is tolerated and shaped, so that it might be contained.
And I am offended that your impossible goal is so strong a thread in our ethical tradition. To me, it betrays a deep, patriarchal Statism based on basic internal contradictions that any clear-eyed ethical thinker should see through.
That entire inclination in Western thought hits me as simply classist/sexist dismissal of the working-class/male role outright. Marcuse aside, we cannot universally adopt the middle-class/female privilege of choosing to put aside aggression and presuming someone else will coordinate protection.
By shifting the duty to protect us up to some idealized proxy Father in the Sky or the Statehouse, we have not eliminated the violence involved, or even reduced it. We have simply concentrated it, and we either encourage specialization in violence, which encourages abuse, or we force a greater duty upon the unwilling.