While I cannot speak for the physics of birds, I can say that an engine WILL degrade rapidly if shot prior to take off. Landing gear does not affect this, per se, but will affect how much speed you can gain and altitude.
The issue was your engine was shot, and the stress of flight paired with the damage from the bullets caused the rapid degradation of your engines. It does seem that simple wear and tear can have an affect, though I have not seen an engine fail just general flying if it had not been damaged first. I have seen and tested (by shooting the craft prior to take off) the secondary engine fail due to the stress of being the only functioning engine, after the shot engine failed.
First the left would go (left if looking at it from behind, and the one I'd shoot). It would spark, shoot smoke (it would degrade and fail quicker the greater the ascent angle) and then eventually die. Not long after, the right engine would begin to emit smoke, spark and then also fail, leaving me gliding (pretty fun).
So what is happening is the force generated by the speed and rapid ascent is causing your damaged engine(s) to fail. I've damaged an aircraft and only did the minimal amount necessary to take off and you will notice a clear difference in the speed at which the engine fails. This is using a pump shotgun, three shots into the engine every time.
Think of it kind of like a space craft. Even foam can be dangerous at the speeds they are traveling, just like a loose wire. Same goes for a plane with a damaged engine.