You have basically answered your own question. They killed because they had been “indoctrinated with all kinds of reasons why the killing should be done.” In a nutshell, that’s it.
Sure. Certainly, plenty of “ordinary Joes” in Germany (not to mention Poland, the Baltic countries, Belorussia, Ukraine, Russia etc., etc.) believed, for a number of reasons, that the Jews deserved to die - or, at the very least, that the Jews were a “problem” that needed to be solved, one way or the other. In Germany, Hitler had been in power since 1933, and the Nazi faithful had been lapping up his poison since long before that. By 1941, you had lots and lots of 20-30-year-olds who had basically grown up on the stuff.
But yes - I, too, imagine that not all of them found it an easy thing to kill thousands upon thousands of poor, starving, unarmed men, women and children, begging for mercy. That’s where peer pressure comes in, and threats, and further indoctrination, and all manner of mental gymnastics to short-circuit one’s conscience, decency and humanity. In his Posen speech, Himmler spoke of the extermination of the Jewish people as something his audience - i.e. the appointed butcherers themselves, charged with the task at hand - had to “endure” (“dies durchgehalten zu haben,” “keiner hat es durchgestanden,” etc.), i.e. a disgusting, unpleasant, thankless, but nevertheless absolutely necessary task. By framing it like that, he tacitly acknowledged that for many men, mass murder does not come naturally - while at the same time underlining the basic point that mass murder was nevertheless their duty. (He himself was said to have kept a copy of the Bhagavadgita in his pocket, for much the same reason - to remind himself of a warrior’s duty.)
These men had plenty of opportunities for “desensitization,” too, through the T4 program, Poland '39, etc., etc., for years before the industrial-scale death camp slaughter really got under way, ca. 1941-1942. Along the way, those opposed to mass killings of civilians - men like Blaskowitz - had been either dismissed or reassigned. Those who remained were men like Heydrich, Eichmann, Jäger, Greiser, etc. - relentless killing machines, fanatically dedicated to the task at hand.