I asked a similar question years ago and after a lot of research, here is what I found out.

US Airways wanted to reduce the number of altitude deviation they had. Their SOPs at that time did not address crew coordination when it came to setting the altitude pre-selector. As a result, PF was either forgetting set the altitude pre-selector because they did not hear the clearance or misheard the clearance and set the wrong altitude. They realized that if both crewmembers were included in the altitude setting process, then the likelihood of an incorrect altitude being set would greatly decrease. To include both crewmembers in the process, they had the PM set the altitude pre-selector and had the PF confirm the altitude. The rest of the industry began to adopt this procedure since it proved successful in reducing the number of altitude deviations.

During the mid-2000s however, arrivals with multiple crossing restrictions became more prevalent. Altitude deviations began to increase and studies showed that PMs were setting the altitude pre-selector WHILE they read back the clearance to ATC. The practice of setting the altitude pre-selector WHILE they read the clearance back prevented them from forgetting the altitude assignment. This became an issue when the autopilot was in the process of capturing an altitude. When the new altitude was set before the current altitude was fully captured, the aircraft would enter pitch hold mode and would not actually level off. When issued a crossing restriction that allowed crews to delay their descent, the PM would end up setting the crossing restriction altitude right away not knowing if the PF wanted to level off at in intermediate altitude before making the crossing restriction. Operators began to understand that similar to the heading knob, the altitude pre-selector actually controlled the flight path of the aircraft and should be in the control of the PF to prevent unintentional autopilot mode changes.

Airlines began to change their procedures back to the original method of having the PF set the altitude pre-selector BUT now included the PM in verifying the set altitude. They realized that the initial decrease in altitude deviations US Airways experienced wasn’t due to the fact that the was PM physically setting the altitude pre-selector; it was because BOTH crewmembers were talking about the altitude assignment. As long as both crewmembers verbalize the assigned altitude and verify that it is set in the altitude pre-selector, the person who sets the altitude pre-selector does not matter.