The no assigned seating is a remnant of the PanAm shuttle, which Delta bought during PanAm's death agony. My first flight on a DCA-LGA shuttle was in 1977, when I was a messenger for the Washington office of a New York law firm. This was the famous Eastern Air Shuttle, where there were no first class seats, no food or drink, and you could board right up to push back time; if you were there on time, and the flight was full, EA would roll out a second section that would leave 15 minutes later. You paid on board, the FAs rolled carts from front to back to take your payment by cash or credit card (I think the fare was $75 on weekdays back then). My recollection of the trip was that I was given a pre-paid voucher by the firm, which I gave to the FA on board; you filled out your own boarding pass before passing through the metal detectors. In fact, I didn't have to put out any of my own money for the trip, as the firm arranged for a car to pick me up outside LGA (EA used the main terminal; PA, which was the successor to the Frank Lorenzo-founded New York Air Shuttle, used the Marine Terminal), and put me in a cab where they had an account. I was a 22 year old kid, and, let me tell you, it was exciting at the time.

The reason it takes 37 minutes, as opposed to 20-25, is that the route from DCA to LGA is one big zig-zag. Assuming the flight takes off from south to north, the plane follows the Potomac River to the Beltway (American Legion Bridge), then turns right and flies to the Chesapeake Bay Bridge, then turns north up the bay, then turns right at the Chesapeake & Delaware Canal, then turns left at the Delaware River, flies overhead PHL and then to a point somewhere over central NJ near the Raritan River, then turns east, then enters the pattern to land at LGA, which could mean circling NYC to the northeast and then landing to the southwest.