I've read accounts by people like Franz Suchomel where they state that there was no real volunteering or selection process - one day they got orders to report to a new secret program and that was that.
Are these correct? I can see why, if they had volunteered for genocidal duties, they would wish to hide that fact. Was thought ever given by the higher-ups to what kind of person they wanted to carry out this work? Personalities like Christian Wirth or Erich Bauer seem to ideal in that they were clearly unhinged psychopaths, whilst the likes of Suchomel and Groening present themselves as Browning's "ordinary men" who simply followed orders without engaging their own consciences.
This is a complicated question. In general, the Nazi KL system was founded by Theodore Eicke. He tested a lot of the practices that would be commonplace in the KL system, most notoriously the "Kapo" system that set prisoners against each other, by issuing certain inmates privileges and powers over their fellows, making them targets that other inmates could turn on.
One of his "students" was Rudolf Hoess, who would eventually go on to found Auschwitz.
You mentioned Christian Wirth. He and the Aktion Reinhard staff were a very special case. Simply put, most of the Aktion Reinhard personnel like Wirth were drawn from the T4 euthanasia program. They had experience in dealing with killing large groups of people, so Odilo Globocknik - the founder of Aktion Reinhard - handpicked them and had them apply that experience to The AR extermination camps.
Outside of them, there was no real selection procedure. The KLs were run by the SS, so SS personnel were transferred or assigned to the Camps as needed. They weren't naturally "psychopaths", but gradually adapted to the situation of the camps, depending on their own personalities. Treblinka was a pure extermination camp, so the SS acted accordingly and relished the power they had over the victims, often abusing them as they got off the trains and were driven to the Gas Chambers. Groening, an accountant with no real power in the process, chose to try and ignore it, and atoned as best as he could by testifying to what he had seen and experienced.