Majority of Jews killed in the Holocaust were actually shot, or killed by forced labour, hunger, cold and diseases, close to where they had lived or been found, not transported hundreds of kilometeres to death camps like Auschwitz. Extermination camps didn't really operate until 1942, while Einsatzgruppen and their local collaborators (see Schutzmannschaft or Arajs Commando) operated from 1939/1941 (this date varies for different collaborating groups). One of the most drastic examples of such operation is the Babi Yar massacre, where 30,000 Ukrainian Jews were murdered over two days in September 1941.

Those killed in the extermination camps were mostly Jews from what used to be western Poland, and other countries west of the former Ribbentrop-Molotov line, which was boarder between Third Reich and USSR between 1393 and 1941. One of the reasons they were transported to the camps far away, instead of being killed in France or Hungary, was that these countries were still recognized as states by the Germany, and thus their citizens had some legal protection, even when they were Jewish and their governments collaborated with Third Reich to various extent. This way Jews from fascist Italy or Vichy France paradoxically had much bigger chance to survive than Jews from Poland, government of which never cooperated with the Germans. Poland and USSR were not considered by the nazis to be states, so its territory became no-man's land and their citizens no-man's people, that you can do whatever yo want with. At the same time government of occupied Denmark refused to put its Jewish citizens to the ghettos and actively support their escape abroad, and there were no major repercussions from the German side. In Ukraine and Lithuania there was no government to reject nazi demands. By deportation of Jews from France, Germany or Hungary to territories that used to be Poland, nazis made them no-man's people.

To sum up, over 2 million European Jews were killed by Einsatzgruppen and their collaborators, 2-2,5 million died in extermination camps (not all of them in gas chambers), and about 1,5 million were died in other circumstances, such as ghetto pacifications, pogroms, hunger or simply found and shot on the spot in the woods by regular German soldiers. Reason why Auschwitz and other extermination camps are seen as Holocaust symbols (I guess that made you have this assumption) is not because the majority of the victims were actually murded there but rather that so many people were killed in this particular places, in such industrial way.

Source: Timothy D. Snyder, Black Earth: The Holocaust as History and Warning, 2015

Read Bloodlands: Europe Between Hitler and Stalin by Timothy Snyder. It's an exhaustive chronicle of the extermination programs of each respective regime, and argues quite persuasively that the development of extermination camps was a direct result of the combination of the two factors you mentioned in your post.

Yes, at first they did shoot the Jews. The 'Endlösung' (Final solution for the problem of the Jews) went in stages. In 1942 Hitler wanted to start with the final stage and just shooting the Jews did not go fast enough, so they came up with this industrial way of killing the Jews faster.