That does match with the rest of the story. Left turn of of 18/19 to go up river would point him back towards the Whitehouse and Capitol. It does make their mistake a little more egregious though because if they're 'following the river' turning nearly 180 to do that and also not following the flow is much larger change than simply following your current heading and making the small turn to follow the river out to sea. That unexpected and uninstructed turn back towards the WH and Capitol certainly would have raised a lot of eyebrows.

One thing I am having trouble with is picturing where they would have been crossing the river at 1500 feet. The Potomac is basically at the end of the runway he took off from.

On the other hand for the rest of the process small planes aren't that rare in National and there's even a rather scenic approach pattern that actually follows the Potomac into runway 19. https://www.youtube.com/watch?v=Zco3XlYt6Ko

During flight school, one of the first things we learn on a piston-single aircraft is to apply right rudder and maintain runway heading on takeoff. Small single-engine planes have a tendency to yaw left due to effects of torque, slipstream, and p-factor. I'm guessing the pilot drifted left on climbout due to lack of right rudder. He could have turned 90 degrees by the time he leveled out at 1500 feet and been over the river. Sometimes as pilots, the worst thing we can do is react. It's usually best to take a breath, think about your situation, cross-check, decide, and act intentionally. By rolling the dice, the pilot had a 50/50 chance of making the wrong decision. I don't like those odds.