Is it normally the aircraft manufacturer’s responsibility to train pilots, or do the airlines usually handle it? And does that vary by country, etc.? Just asking...
I guess I'm what some have called a Boeing Apologist, but Boeing had a contract with Southwest that the 737 MAX would only require a few days of online "familiarization" training with no simulator time for Southwest's pilots in order to be certified to fly the MAX. Otherwise, Boeing would owe millions in compensation to Southwest if their pilots needed simulation training to transition to the MAX.
It was because of this commitment to Southwest that MCAS was put into the aircraft in the first place. Because of the larger engine nacelles placed slightly more forward under the wing than the older engine nacelles, when the aircraft was in a high angle of attack at low speed the extra surface area in front of the wing caused the aircraft to suffer a pitchup motion just before stalling. MCAS was to provide nose down trim during these low speed, high angle of attack maneuvers, so the aircraft behaved more like the previous 737 NG.
If it weren't for the potential contract penalties, there would have been no MCAS. Unfortunately, Boeing's implementation of MCAS was flawed in that it didn't cross-check both AOA sensors, but instead just relied on the input from one AOA sensor to make its MCAS decisions. If the AOA sensor was faulty, the MCAS inputs were faulty.
A further flaw was that the Boeing test pilots felt that more MCAS authority was required, so the MCAS system was allowed to provide repeated down trim commands, instead of just a couple of down trims per incident as originally designed. Boeing also never told the FAA that they increased the MCAS authority so much, so the FAA approved the more limited version of the MCAS system without requiring pilot simulator training.
But even with the potential for MCAS failure, it results in a condition called "pitch trim runaway" that all 737 pilots are supposed to be trained in and should be able to identify and correct immediately. It is a "memory" item, which is a procedure that must be memorized without having to refer to a checklist. Regardless of whether the runaway pitch trim was caused by a faulty trim switch or a faulty MCAS, the pilots should have been able to immediately recognize the issue and quickly disable the electric trim system.
After the Lion Air crash, Boeing issued an emergency technical bulletin that runaway pitch trim caused the Lion Air crash, and reminded all operators about the "runaway trim" procedures and required all pilots to review the required steps. There no excuse whatsoever for the Ethiopian Airways crash. They were fully informed about the problem, were told exactly how to counter it if it occurred, but they explicitly ignored the correct runaway procedures and kept re-energizing the electric elevator trim system instead of reverting to the manual trim wheel.
Boeing's technical bulletin issued after the Lion Air crash: https://lbblawyers.com/wp-content/uploads/2019/03/Boeing-Service-Bulletin.pdf
If the Ethiopian Airways flight crew heeded the above bulletin, there would not have been a crash.
This came out of the tragic case of the late New York Yankees catcher Thurman Munson, who lost his life in the 1979 crash of his private jet that he probably had no business flying. Cessna had a deal where they sold private aircraft to customers and included the training/licensing of the aircraft as part of the purchase contract. In the aftermath of Munson's death, his family sued Cessna on the grounds that this arrangement represented an obvious conflict of interest on the part of the company -- namely, that the company has a financial incentive to give the pilot a license regardless of his ability to fly safely, just to complete the sale of the aircraft.
I seem to remember reading that this practice of having manufacturers train and license pilots was discontinued some time after that incident.