There’s only one person that makes the final decision. Me. Any airline that forced the issue would be on very thin ice legally both from a straight authorisation point of view and from an employment law POV if they hoofed the pilot. Apart from anything else the Air Navigation Order and our Ops Manuals are explicitly clear on my primary role; the safe conduct of the flight.
In making that decision I’d seek guidance from Engineering and ask the view of my oppo. If either have any doubts, there is no doubt. I’m firmly of the opinion that being around to be told I was wrong is a better option than not being around.
I’ve actually been in the situation where I’ve binned it based on the best info I could get and was subsequently found to have called it wrong. I got given some tech feedback which was useful knowledge gained but the only feedback I got on my decision was “wrong call but for the right reasons, keep it up”. Long may that continue.
I can’t speak for other operators but in the developed world there are varying levels of subtle pressure applied but I’d very much doubt anyone would get disciplined for it. Any Captain that folded shouldn’t be in the left seat anyway.
I get @CrashTestDummy ’s point about being the first up but again, it’s a personal call and re the Max, my management are under no illusion that many will take some convincing. As for someone ordering you to get in a Lynx and fly following the death of friends, that shows an appalling lack of awareness on a man management level and IMHO, a disgraceful abuse of authority worthy of Court Martial at the command level. That said, I’m wholly unfamiliar with the incident in question.