That is a fair point. I guess it's important to distinguish with the term "Expertise". I viewed expertise as someone proficient in killing humans, i.e. serial killer, trained military, etc. The proficiency in taking large game does have some correllation as you mentioned.
Most hunters when they kill a big game animal get "buck fever". It's a body's real response to seeing and preparing to take game. Hands shake, heart races, adrenaline pumps. Hunters can combat these effects with Beta blockers, or anxiety meds, but the natural body response does still exist.Killing 1 person can happen in a "daze" so to speak. Immediately after, some of the adrenaline wears off and the person snaps back to reality and often panics. This panic usually leads to the killer fleeing, but I guess it could lead to "tying up loose ends". To kill one person and then move on to 3 additional without "snapping out of it", in my opinion, speaks to a highly motivated individual, fueled by hatred, scorn, fear, etc.
For killing humans, anyone that's doing it for the first time will experience similar responses described above, but they're often desensitized as it happens more and they learn to effectively rationalize and compartmentalize their emotions. I speak to this based on experiences with guys with whom I served in the military.