This paper is not about machine learning. "Training" has nothing to do with the approach. External cameras are used because this paper is about trajectory generation and not about vision.
The paper presents an approach of generating a time-optimal trajectory through waypoints given physical limitations of the underactuated system. This is interesting and novel, and as demonstrated works very well. The group from which they come also work a lot with high-speed machine vision, and one of the next research steps will be combining this trajectory generation algorithm with onboard computer vision.
On page 32 of the paper they provide rankings of the two pilots against whom the automated system flew (Michael Isler and Timothy Trowbridge). My personal opinion of course, but I would call both of these pilots "serious human competitors" given both have been competing since 2017 in international events and received many podium finishes. But that's also beside the point.
This paper is about generating time-optimal trajectories through waypoints given the system's physical constraints (e.g. limitations in thrust and rotational rates). A time-optimal trajectory is a trajectory which is time optimal—meaning that no faster trajectory exists. Given that this algorithm generates the fastest possible trajectory through the waypoints given the physical constraints of the system, it would be impossible for even a "serious" human competitor to beat it.
You are incorrect. Generating trajectories is easy. Many well known techniques exist that do pretty well, and yes this is done in computer games all the time (as well as in many other fields).
Quickly generating time-optimal trajectories for under-actuated mechanical systems with actuator constraints is interesting, and as a researcher in this field I can assure you that the technique in this paper is novel and is interesting—if it were not it wouldn't have been published in the journal Science...
You are correct, if you constantly hold it offset from its balance point, the wheels will spin up until they saturate and it falls over.
The overshooting of the balance point isn't intentional, but a byproduct of the "learning control" employed to improve the jump. Brake residue alters the stopping ability of the brakes (and thus the duration of the energy transfer into the cube's body), and a slightly offset center of mass affects the location of the balance point. In successive attempts at jumping up the energy of the wheels is altered to compensate for these effects, if it undershoots once, likely it will overshoot the next time as it "hones in" on the correct settings. Once the jump is close enough to the balance point, the control switches from open-loop "jump" into closed-loop "balance" to stabilize the cube around the balance point.
The wheels are actively controlled by the motors, so it's not gravity that is slowing the wheels down, rather the moment that is required to stabilize the cube's body. There is also friction in the system, so some energy is always lost and thus the wheel rotational rates will tend to zero around the balance point.
The paper presents an approach of generating a time-optimal trajectory through waypoints given physical limitations of the underactuated system. This is interesting and novel, and as demonstrated works very well. The group from which they come also work a lot with high-speed machine vision, and one of the next research steps will be combining this trajectory generation algorithm with onboard computer vision.