Why not it just display everyone's place?
Corresponding to what, the checkpoint? If this is done, it's inaccurate, because there's no way of telling how the places have changed in between checkpoints. If you suggest waiting for everyone to cross the finish line, I feel it'd be better not to wait to see last place continue being last place.
On the other hand, I thought up a small system in mind that could work for genuinely live placements, but I don't think it'd be efficient. The idea behind this system is to have the map maker place invisible beacons that, as a player gets closer and closer to the next one, it means the place is advancing, providing the player with a real live placement in the race. The problem again is basically the large amount of checking that needs to be done. Another problem could possibly be split paths, I have no idea how to handle that as placements would no longer be live (Possibly update as soon as the reach contact with a new beacon if their at the right checkpoint level). In anycase, it's still something currently in consideration but this current list are the plans for now.