That would seem to fall directly on the operator / carrier.
Boeing takes responsibility if the training manuals were not comprehensive. Aviation authorities take responsibility if they under-mandated training. But ultimately neither know a given carrier's scheduling and pilot allocation.
If the carrier was told by Boeing and the FAA that no additional training was needed, why would they use any different procedure.
At some point this falls back on nannyism. Either you have a functional company, where your employees are empowered to communicate internally, and your company does something about it... or you don't.
Boeing's job is to engineer the billion things required to keep an airplane flying. The government's job is to validate Boeing performed due diligence and mandate what training is required to the carriers. The carriers have a job too.
I wouldn't be surprised if Boeing pushed hard to have it same type certified. Why wouldn't they? Ultimately, regulatory agencies agreed. Maybe that was correct, maybe not.
But absolving the pilots' direct parent company of responsibility is bullshit. Aerospace is a different industry, but it's not that different vis what a healthy, functioning company does.
It's a damning indictment of Boeing, and whichever authorities let them get away with it, that they pushed through an aircraft with a different pilot UI/UX knowing that pilots would then be under trained when using it: like someone said we train baristas better.
I've driven minibuses before and always ensure to get an hour of driving a new model before taking passengers.