Even the stuff that's built to rigorous standards is mostly built with a serials of failsafes because the same issues of being able to trust the thing apply. But it's fine because like you note, failsafes limit the effect of any individual failure.
Beyond that too, I think it's just important for people to realize that most things just don't matter that much. Everything we do eventually has to reach human scale and at human scale, there's so much room for slop and correction that most failures don't matter. Missing a paycheck because the payroll computer dropped your row is awful, but society can be built around allowing people to recover from that. Late payment forgiveness, forbearance, procedures around cutting a new check early.
I think the biggest danger about the "presumption that computers are working correctly" is in it's ability to erode our understand of the fact that "to err is human". It's too easy if you assume the machines are perfect to assume that any errors are therefore malicious, and so it's easy to go from a system where you have "oops my tax software messed up, no big deal I'll file an amended return" to "the only way for a mistake like this to happen is to intentionally put in bad data, so you must be guilty of tax fraud". Or even worse "the computer won't let me do it". I understand you want computers to have check in place that prevent the overriding of procedure and you don't want to give your low level employees the ability to override procedure at will. I also think that EVERY system should be built such that it is possible with a certain combination of people on site at the point of contact to override ANY check in the system to make the state of the system match what it should be. My specific example in this case was a problem with a prescription at the pharmacy. The script had been partially filled because they had stocking problems, but instead of going into the "partial" state, it was closed as filled completely. The tech knew what had happened and knew we still were owed the rest of the script. The pharmacist knew what had happened and agreed we were owed the rest of the script. The store manager agreed. And yet no one could change the state of the system. The only option was to get a new script from the doctor for a new fill entirely. That's the danger of assuming everything is working correctly all the time.