I agree with you that 'self-ownership' isn't a helpful term. Ambiguous, confusing.
When the argument is made that idea of property has its basis in so called "self-ownership," an abstraction is created from the right to defend one's personal integrity. That single aspect of the complex relationship between me and my self is divorced of its context and applied to external objects.
I disagree that there's anything being divorced from it's proper context here though. When you say 'defend personal integrity' the only thing that can intelligibly mean to me is defend my property, in this case my body.
Why is it okay for a doctor to cut me open but not okay for a stranger to spit in my face?
It's because I've consented to the doctor using my property in that way, I have not consented to the stranger doing so.
Clearly you have the right to defend your brother if he is being attacked.
Yes. And we differentiate 'attack' from other forms of interaction, in part, by referring to property rights and judging whether they're being violated or not.
But you don't own him, and it would be a rather silly argument to make that you do.
Of course. Your right to defend him is a specific example of your right to use force to prevent any rights violation from occurring. In this case the violation of your brother's property rights in his body.
Since the concept isn't necessary to justify defending another person
It is necessary. The concept of property rights is necessary in order to determine whether the use of force is justified. Consider: Your brother's arm is amputated and he's not expressed a clear wish to keep it (perhaps he fell unconscious) - his ownership status over his arm is now uncertain. This does have an affect on whether it would be justified to use force to prevent someone else using that arm without your brothers consent - now it's much less clear whether that use of force would be justified, because (I claim) the ownership claim of your brother is much less clear.
We disagree.