And why would Lee just go sit and have a beer after being shot in the chest?
I actually thought this was the best assassination in the series. As Lee was losing blood, he probably came to the realization that fleeing was only a temporary solution. Connor would continue to hunt him, and find him, and succeed in killing him just like the rest of the Templars.
That final kill is symbolic of the tenacity of idealism. The Templar Order is concerned with control over base passions, whereas the assassins, and particularly Connor, use their zealotry toward the cause of liberty (and revenge) as a tool.
Lee has ideals, but his have been pushed on him by his order. Connor's genuine spirit makes him a much more dangerous foe.
I didn't love everything about this game, but I did think that scene was atypically poignant for Ubisoft.