The history of the death of the Jews, the Roma, homosexuals, the mentally challenged, Slavs, and other untermenschen by the Nazi regime is available. Leni Yahil's massive The Holocaust details the gradual development of the policy, ending with the killing centers in Eastern Europe, but not flinching over the important role of the Einsatzgruppen in the first days of the war, as special SS units moved in to Nazi-occupied Poland, rounded up entire communities, and killed them via firing-squad. The mobile killing units later included specially designed trucks which would pump poison gas into trailers. These were the seed-bed of the gas chambers, yet they are also responsible for the deaths of thousands.
Auschwitz as a short-hand for the Holocaust is indeed inadequate, because the event, like all historical events, is a complex, developing creature. Auschwitz is the full-flowered, final development of a national policy geared toward eliminating surplus populations, while attempting to gain as much economic advantage from at least some of them as possible.
I believe Snyder is correct that our understanding was limited, at least in the beginning, due to the evidence. We had survivor memoirs, especially Elie Wiesel's. We did not have, for years, a complete grasp of the enormity of the policy, a grasp of the development from legal disenfranchisement, the various attempts at a "solution" to "the Jewish question", and the various related policies for eliminating surplus populations that eventually became incorporated into the systematic murder of whole populations.
Snyder is also correct that the role of the Soviet Union, its mass murder of Jews, Ukranians (especially Ukranian collaborators with the Nazis), and others during this period is little understood. Unlike the Germans, however, the Soviets were both victors and allies, their records off-limits to independent historians for study. The whole story, for example, of the Soviet Army executing most of the senior Polish army in the Katyn Forest, has only been told in full in the past couple decades, yet, as Snyder notes, this is only the tip of the iceberg. Indeed, the post-war mass murders of Soviet soldiers held as POWs by the Nazis has yet, to my knowledge, to be adequately addressed (that it happened, is pretty well known; the why's and how's and wherefore's, however, are still hanging out there).
We need to escape from the easy shorthand of symbols. Indeed, as far as I'm concerned, symbols should be eliminated entirely from our ways of grasping the world around us. Flags are just pieces of cloth. "Coke" is a consumer product, not shorthand for any carbonated beverage. "McDonalds" is a franchise restaurant, not a symbol for American hegemony. Auschwitz was a horrific place where millions died, yet it was only one of several killing centers in Europe, and hardly the only place the Nazis killed people; it was not, in fact, representative of the multi-yeared, multi-layered structure of the development of the Nazi genocide. At one time, it might have served well as a symbol. Our greater understanding of the entire phenomenon, however, renders it inadequate as a symbol. Rather than seek out new symbols, it might be important for historians to communicate the whole, terrible, complex reality, as fearsome a task as that may be.
We owe it to the victims, and to ourselves.
UPDATE: Understanding is not helped by bs like this:
“The truth,” wrote Dershowitz, “is that the Palestinian leadership, supported by the Palestinian masses, played a significant role in Hitler’s Holocaust.”
"Significant"? How many Palestinians were guards at killing centers? Were there Palestinian units in the SS? This isn't just bizarre, it's crazy-stupid. Alan Dershowitz has lost any credibility whatsoever.