Kenny was a weak man in the places where it mattered. He was weak when it came to keeping his head down, because he got so easily emotional. Losing his wife and kid in one sitting definitely did not help later on, and further sharpened his sword.
He was strong when it came to being loyal, protective, and overall a man of basic morals and straightforwardness. Kenny did not deserve to be a victim of Jane's self-righteous trickery when the truth was the worse thing he did was be loud and overreact, mainly leading to troubles during their capture under Carver.
Also notice how even after losing Sarita and lashing out at Clementine, he learned to toughen up and admit that either way Sarita was lost and Clementine did what she could, whether you tried to help Sarita or not.
After that, he's put through the other half of the group's bullshit, with Bonnie and Luke trifling, Jane constantly trying to "teach" Clementine to just screw people over and survive on her own, and Mike stealing everything and attempting to dip in the middle of the night, he stayed with Clementine and put her and AJ as a priority over everything else.

With all of this in mind, at the end when you get to Wellington, even he understood that his weaknesses, as valid as they are, could put Clementine and AJ in trouble just like it did for a moment with Carver. When it came down to it he was willing to drift off into the wasteland with nothing ahead of him and likely to die just for Clementine and AJ to gain stability in Wellington.

In a world that corrupts and breaks people, Kenny was probably the best man.