So, your last question about this growing out of localized mass killing into a bigger program, so to speak, is pretty close on the money in some ways but there is a couple of crucial factors that are paramount here:

The first one is, that what the Nazis did with Reinhard nobody else had done before, at least not on such a scale and with such sytematicity, simply because nobody at that point had had the resources of the modern state on their hand and pursued such a goal before. To us, we see those decisions in hindsight and thus can see a clear path emerging that goes into a specific direction but to the historical actors in the Nazi government, this was all uncharted territory. The violence of the Einsatzgruppen was violence that resembled in some ways a system of colonial violence, if you will. Taking groups ostensibly charged with security going to a place and executing a number of people they view as "risks" or as foreign or the other is a system of planning, of logistics, of practice that was not that complex to execute, because there had been precedent for it and it was part of what Enzo Traverso would call the "colonial archive" – meaning a system of knowledge familiar to European powers about how to deal with a colonial population. But something like Reinhard and later Auschwitz required a lot of parts coming together and the Nazi administration both on the ground and in Berlin and between had to figure out and establish. And to use it seems like a straight and logical development seen in hindsight but for those putting it together it was far from that, especially since there are a lot more factors than "just" the exterminatory goal figured into it.

And this leads to the second point: That there were at least two larger issues that heavily figured into and shaped Aktion Reinhard. The lager Germanization plans of Poland and the issue of providing food to the German populace.

The name "Aktion Reinhard" was given to the program of killing under the unified command structure of SS and Police Leader Lublin Odilo Globocnik. However, the construction of the Blezec camp starting in November 1941, which would become an integral part of Aktion Reinhard was well underway before the name and the program had been developed as such. This was due to the fact that it found its origin in another program, the so-called Programm Heinrich. Programm Heinrich was developed by Himmler after the German invasion of the Soviet Union and intended to be a concerted effort to Germanize certain parts of Poland.

Modeled on the Generalplan Ost – which I go into here – Programm Heinrich foresaw a new form of German colonization in Poland, including German villages, settlers and so forth, closely modeled on some forms of settler colonialism as well as a sort of mythical model of "Wehrbauern" (farmers who would double as soldiers defending the frontier). To this end, Himmler tasked Globocnik with preparations for such an undertaking in July 1941. As envisioned, it included massive resettlement and murder of the Polish population as well as ridding the area of Jews.

The General Government and the Polish territories in general at the same time were also the place where the Nazi leadership encouraged leaders from other territories to send their Jews to, together with the Soviet territories. And when it became blatantly obvious that a.) neither would there be a further dumping ground for Jews further East because the Soviets weren't beaten yet, b.) Hitler ordered the deportation of Jews from Germany to the East in September 1941 and c.) the threshold to wholesale mass murder had been crossed in the USSR in the summer of 1941, the local actors understood that these programs of Germanization of Poland would include the mass murder of the Jews in those areas.

It is exactly that around this time, the first two death camps start to be build: Chelmno near Lodz and Belzec with only the latter being under Globocnik's administrative structure. It's also clear from the capacity of these camps that they were most likely not intended for all of Europe's Jews but rather for local killing actions, namely to kill the Jews of Lodz in Chelmno and the Jews of the Lublin area in Belzec.

Chelmno was set up by Herbert Lange – on who's orders is unfortunately difficult to reconstruct –, an SS man and most likely engineer of the first gas van that he had used to murder the inmates of asylums in Poland (also connected to Germanization policies. Belzec was staffed by men who had collected massive experience in mass killing through their involvement as "experts" in the T4 killing program in Germany and who would go on to staff the Reinhard camps for the remainder of their existence. And both camps were comparatively small in their capacity to what was later built. Chelmno only basically consisted of a local base in a castle and a gas van, while both crematoria and gas chamber capacity was noticeable smaller in Belzec than in the two other later Reinhard Camps.

The plans for large-scale Germanization of Poland (and the USSR) however largely turned out to be impossible to pull off under the conditions of the war and were by the end of 1941 and beginning of 1942 if not abandoned then postponed to after the war, save for a few exceptions in some areas. However, in some cases such as Chelmno the camp had already started operating and in Berlin around the time of December 1941 the decision for a Europe-wide program to murder all Jews was taken. This most likely entailed the idea of using the already existing local infrastructure like Chelmno and Belzec to that end.

But as we know from the protocols of the Wannsee Conference, the plan was originally to comb through Europe West to East in arresting, concentrating and deporting Jews. In history, however, it happened pretty much the opposite way with the Reinhard Camps starting to operate in late/spring Summer 1942 and killing the Jewish population of Poland through out the next 7-11 months. The most likely reason for a shift in this geographic focus was related to the issue of food planning.

Food was for many in the Nazi leadership a pivotal issue because of their experiences in WWI where food shortages had been a major factor in ending the war. And when due to the invasion of the Soviet Union and Ukraine not turning out the breadbasket they'd hope it would be as well as the GPO revealing itself as unrealisitc, it is very likely that the decision was taken to speed up the planned killing of Polish Jewry in the Reinhard Camps.

In short, the Reinhard Camps originated from a process that was not planned out from the beginning but changed and shifted in purpose and scope from the beginning and so, this was also reflected in the structure wherein several camps were build with Sobibor doubling for the same area as Belzec and Treblinka serving a different district (this is also reflected in the death tolls, e.g. that Treblinka killed more people than Belzec and Sobibor together). Rather than extend the capacities of one camp, it was just decided to build another camp.

So, in the end, it did indeed grow out of the localized operations and then expanded over a time when policy goals shifted. The same way, it was closed down when the Polish Jews were dead about 11 months later. The structure and capacity simply wasn't needed anymore and in an internal policy conflict, Höss as commandant of Auschwitz had been able to position himself against Globocnik and positioned Auschwitz as the central camp, where about as much people as in the Reinhard Camps were killed.