Did the FO even check the FMC entries before departure I wonder? And why did the skipper manually enter the lat/long into the FMC from the sign at the gate? At my company we just copy the GPS coordinates and verify them against the airport/gate reference from the database or JEPS if required. I have stuffed up lat/long waypoint entry before on a long sector with about 10 manual waypoints to enter because ACARS didn't download them, but this is why the second crew member checks the FMC. The error was detected during the LEGS check and it turns out I had some finger trouble during entry and the waypoint was about 50 miles off track
very stupid on my behalf and I learned a valuable lesson that day, now I go over my own work with a fine tooth comb before passing it over to the other guy. Having said that, departing with an incorrect alignment in this day and age should be impossible and ignoring all of the warnings is just insanity
