The airline will fix the bug and it won't happen again. But relying on a closed, unauditable platform to keep any sort of safety-critical data in these conditions seems absolutely unconvincing to me.
Furthermore, you have 2 iPads in the cockpit, so it could only be a problem if both fail at the same time. This seems like it'd be a more rare occurrence in the safety case.
In terms of whether open source could have helped, I doubt it very much. Stopping a bug like this (beyond extensive in-house testing) is to largely declare the platform and software bug-free (individual audits typically focus on properties like security, which are easy to target), and a full operating system (including UI) is far too large to effectively audit in that way. Certainly iOS gets far more user milage than any free alternative, which is a big advantage there.
I'd have far more faith in Apple's display system than I would the eminently auditable Gnome (this is all assuming that the FAA did not have access to the iOS source, which is unknown at this time).