One drawback that I haven't seen mentioned yet, is single-wheel torque. If you look at the video, there is a pretty sizeable motor that's even geared down with a sprocket an chain. So there is a lot of torque and power available, and due to the solid axle, you can send it to a single wheel. This might be important in off-roading situations.
With direct drive motors, you have to make a pretty big compromise somewhere:
* Two smaller motors (same total power): drastically less torque per wheel
* Two larger motors (>2x total power): somewhat less torque per wheel (still don't have gearing!), cost, weight
If you're adamant on avoiding the solid axle, I think a good-ish solution might be hub motors on all four corners. That way you're far less likely to depend on a single wheel to get you moving.
Or alternatively, a locking diff in the rear (forget about LSDs). They usually provide some gear reduction, so you can skip the chain and drive it directly with the electric motor.
In short, the solid axle is a really good choice that you would be hard pressed to replace, without making a sizeable compromise somewhere.