Well, Strike Eagle pilots practice defensive and offensive counter-air just as much as they do interdiction, SEAD, and the like.
One of the reasons F-15E units serve as "on-call" fighter squadrons for short-notice deployment is that mission planners get two for the price of one. Need air superiority? Call the Strike Eagles. Need a capable strike fighter? Call the Strike Eagles.
Navy is going with the Super Hornet for much the same reason, I suppose.
Although with the current acquisitions of the F/A-22 and the Joint Strike Fighter, the Pentagon seems to be going in the other direction again. Guess there is something to be said for specialization.