What you say is only half true. I’m also thinking of injuries caused by animals and other people. Antiseptic isn’t going to fix the nasty kind of infections deep bite or knife wounds cause. A hunter gatherer society is definitely at greater risk of suffering these kinds of injuries than we are.
And also, even antiseptic treatment was in shorter supply than it is today, so it’s still a moot point.
There's sufficient evidence that hunter gatherer societies have indeed used various plant- and animal based antiseptics (honey, oils, tannins, resins, fungi,...) to treat wounds.
I said shorter supply than today, not totally unavailable. Pre-agrarian societies, by definition, were not growing and harvesting antiseptics in bulk. They’d not do much against an infection from a stab wound (yes, non-agrarian societies encountered, fought and killed each other).
And also, even antiseptic treatment was in shorter supply than it is today, so it’s still a moot point.