Excellent!! Not only did you get it working, but you could see exactly why it wasn't working. Techs like to achieve both answers and even the next step 'how did it get into a faulty situation in the first place'.
From my experience, I can say that that with regard to 'how' and as you quite rightly noted, one has to be careful with pin alignment and proceed with caution and sensitive touch.
With many male/female connectors of this type where the connectors are sitting inside cavity's of the plastic/bakelite housing, the wire/s may be tensioning the pins a little one way or the other and with the clearance between the pin and cavity walls, can allow the other end of a pin to be moved off centre far enough such that it doesn't align adequately with it's mating hole and can end up getting bent over.
4 pin molex connectors as used in computers are serial offenders of this problem, especially when trying to plug in an extension/adapter cable.
They're reasonably chunky pins that don't bend easily, but I've been to hell and back on more than one occasion trying to complete a mundane/routine task as connecting an adapter cable.
It was lucky that no pins were broken off in your cars connector.
. Straightening them back again can require even more of a delicate touch.