In the 1980s and 1990s the NASL and MLS used an unusual kind of penalty shoot-out to decide drawn games.
The system used was like that used in hockey: starting from 35 yards out, an attacking player was given the opportunity to dribble the ball toward the goal and had five seconds to shoot. The goalkeeper was able to run out to prevent the shot as they would in a one-on-one in normal play. See, for example this youtube clip.
In the MLS in this period all games had to have a winner, so the system was routinely used to decide games that were drawn at full time.
Why was this format discontinued? Was it a technical deficiency in the format or was it a matter of adopting IFAB rules more closely (e.g. by allowing drawn games)?