The fact that you have private property by itself is not a sufficient justification for arbitrary force on others.
well, in this sentance you havem issed out a conjunction. as whilst the fact that you have private property by itself if not a sufficient justification for force on others ( though it is necessary)
perhaps the fact that you have private property, and this property is being aggressed upon allows you to summon an arbitrary degree of force against the aggressor
the issue of what is necessary to ‘resolve’ the situation is a diversion from the point at stake, i.e. what actions are to be tolerated, and what actions are disallowed under the context of libertarian law. (when dealing with trespassers, which some would label ‘petty’)
the point that violence wont often be needed to remove tresspassers from lawns, is true regardless of whether landowners have the right to adopt a ‘shoot till they die or leave’ policy, or have no such right. or if the right is to be determined by a coin toss whenever the issue arises.
now, shooting and getting a reputation for shooting can be assumed to bring its own negative consequences that are disincentives from that course of action. but what is it in the action that makes the action incompatible with the non-initiation of aggression ‘axiom’?
it seems to me that all you can say about trespassers is that they trespass till they stop trespassing. and whilst in the ‘trespass’-time-zone, when they are engaged in trespass they have initiated aggression, and anything that is done to them by another agent cant be criticised on the grounds of not-initiating agression. because admittedly the agent is not initiating aggression but meeting an offered aggression with his own.
you have correctly identified good reasons why its against agents best interests to bind themselves and the others in their societies to a code of conduct that positively commits them to violent defence of their property for even the ‘most minor’ offences. And one would hope that a libertarian scoiety of individuals, would by extension of the respect given to each others property, in the shared environment of mutual prosperity would lead to a kindler gentler society, one less likely to go for the gun . Yet this doesnt mean that these individuals do not have the right to violently defend (even if its reasonable to assume they rarely choose to use this right).
the other part of this though is the knowledge problem of judging initiation of aggression, judging that trespass has occurred, this problem is assumed away by many of the purported ‘dilemas’ that simply propose that there definitely is some kind of minor trespass going on.