Exactly, the efficiency had little to do with the method of death, the methods where simply the an example of said efficiency. It's a twisted blueprint. First vilify them, second remove them from anything approaching important positions (collage professors etc), third separate them from society (concentration camps), and finally murder them site unseen. Of course each of those steps where rather complex but the specifics are less important than the overall process. Because, it take a huge number of people willing to do horrible things to create atrocities on that scale.

Arguably step one had been occurring for hundreds of years before the Nazi's rise to power, but they really pushed the envelope.

What I find fascinating is I suspect for a great number of the leaders they really where just looking for something to vilify at the start and things just snowballed. It was only latter in the process as this huge segment of the population because less necessary that the idea of killing them showed up. Something like mass sterilization would have been as effective if there idea really was a twisted form of eugenics and they had been forcibly sterilizing people there and even in the US. However, while the initial idea may have been slave labor they kept sending so many people to these camps that they needed a 'solution'.