I am not a physicist, frame builder or indeed a dirt bike guy so I readily concede I could be wrong ( wouldn't be the first time ).
Your argument does make sense and I agree when designing a bike there are many factors with regard to angles and Tube Length that of course would effect a bikes ability to climb.
When setting my saddle fore and aft and indeed height I always set this in relation to the peddles as I have old knees on there way out, so I would not adjust my saddle to get me further or closer to the bars in order to adjust my cockpit.
So in fairness to the op,the question is simply if I took my bike and shortenend the stem would this make it harder to climb and or keep the wheel on the ground, I am sure you could design a bike with a shorter stem to climb better but would simply shortening or lenghening while all else stay the same make a difference.
My brain indeed tells me it does, but their must be a definitive answer and some one on here must be qualified to give it.