On the contrary, it most definitely did and all sources available indicate that it did. You don't get to handwave away all the evidence just on your say-so.
Utter nonsense. I was discussing what happened to Polish Jews inside Poland - the majority of European Jews under Nazi control lived right next door to the death camps.
Poland was garrisoned by Nazi security and police forces anyhow, because no occupier leaves his lines of communication undefended. The Nazis had 34,000 police in the main part of occupied Poland in 1942, including collaborators. This was more than enough to round up most of the nearly 2 million Jews living there.
They even used recruits from Waffen-SS training units stationed in Poland. You have to station training units somewhere, right? Also meant there were many thousands of trainees who were good enough to point a rifle in the right direction at unarmed civilians.
You cannot honestly be arguing that manpower was a bottleneck here. It wasn't.