I don't think Washington was ever in danger. It was, at the time, the best defended city in the world (I have no citation, but they're out there). I also think that Lee did understand what he was facing, and that was a foe that the south could not defeat in the same way the south could be defeated. Everyone was aware of total superiority of the north in terms of men and equipment. The south had virtually no hope of conquering the north, it had to be a diplomatic defeat through a series of military defeats. He achieved many victories, they north just had enough determination to withstand them. A cataclysmic defeat at Gettysburg may have been enough, maybe not, who knows?
Oh, I don't think Lee won at Antietam. I do think he achieved the best case scenario, though. It should have been a route, though I doubt the end of the war.
Whether or not it would have been an empty victory, though, depended on what a defeated Meade did with the Army of the Potomac afterwords. That's a what-if that we can't know. At any rate, a southern victory at Gettysburg was a long shot with the north's interior lines and defensive positions. It does seem in line with Lee's tactics, most of hist victories were won through offensive tactics (defense of Richmond, 2nd Manassas, Chancellorsville etc)