They need to be good at both, actually. And the left side is hardly the strong side. Statistically teams run at both sides pretty evenly. I've even seen statistics where some teams run to the (offensive) left more often. Regardless, both OLBs (in a 3-4) or DEs (in a 4-3) should be large enought to rush the passer and take on the run.
In the end, Carpenter wasn't exactly impressive on the outside at either stopping the run or rushing the passer. The position requires someone a little bigger and a little stronger.