do we really care about the chicken's motives?

The joke works because we ponder, for just a moment, the possibilities; wonder what reason a chicken might have for making a perilous journey over the blacktop. It's our own gullibility that makes us laugh when we hear the obvious answer, "...to get to the other side."  (Forehead slap!)

Duh. 

Then again...we hear they come home to roost. This implies traveling is in their nature, so perhaps the chicken was on her way home after a long journey. To roost. (???) 

No, we could care less about the "why?" 

The point is everyone has heard about that chicken.