They certainly have a responsibility towards their passengers that goes beyond their relationship with Boeing. Passengers trust airlines with their lives, with Boeing they have 'just' a business relationship.
Airlines have every reason to be skeptical of their supplier even if they do not have the engineering competence to second guess them. They could for instance look through their past communications with the manufacturer and see for themselves which advisories they agree with because for instance they are obviously not safety critical, this would then allow them hire specialists to evaluate the remainder for a second opinion.
Agreed. Among multiple organizations that large and complex, the buck can be passed infinitely. There's the lowly worker who installed the flawed part - the safest target, of course - who can pass it to the worker who made it, who can pass it to engineer, to their manager, back to the engineer who the manager relied on after all, the CAD software developer, to the materials supplier, to the machine tool manufacturer, the HVAC contractor who made the manufacturing facility too humid ...
For almost any act, we rely on other people. That doesn't absolve us of our personal responsibility.
That was in a general case, but in this specific case to satisfy you we can postulate that those on the ground would like to be able to get through their day without having their trajectories intersect with disintegrating aircraft or parts thereof.
Airlines have every reason to be skeptical of their supplier even if they do not have the engineering competence to second guess them. They could for instance look through their past communications with the manufacturer and see for themselves which advisories they agree with because for instance they are obviously not safety critical, this would then allow them hire specialists to evaluate the remainder for a second opinion.