The problem is specific to all tuber type belay devices. The important factor is not the Bergschrund, but the direction of pull in case of a fall. The scenario describe is not falling into the Bergschrund, but down the slope past the belayer. It's the same problem if you belay from your harness in a multi-pitch, that's why you always clip the rope into a quickdraw placed on the belay anchor.
A tube style belay device needs a high, ideally 180° angle between the loaded strand of rope towards the climber and the breaking strand in the belayers hand. That's why the base position is holding your arm down besides your legs when belaying. If there is no redirection of the rope on an anchor/protection between the belayer and the climber, the two strands will end up parallel pointing down, and the breaking force of the tuber will be approximately zero.
Yes, in theory you could hold the breaking strand upwards, but that's against what you normally do (and reflexes matter) - so don't do that.
The same strategy as with multi-pitch climbing as explained above is banned in the article:
The leader's rope does not go through the belay point, as it would be torn out in case of a fall.
Clipping the rope into a biner/quickdraw on the anchor is definitely a bad idea due to the doubling of the forces. However you can totally put the belay directly on the anchor to prevent that. And if you have a solid ice screw or dead man anchor in the snow, that will hold just as much as a bolt (or more) anyway. However again, a tuber wouldn't be suitable due to breaking direction.
Different argument: Unless you need to abseil (long distances), you don't need a tube belay device when mountaineering. Most of the time you wont need to belay, and if you do you can just use a Munter/rock horn/body to belay. Even if you need to abseil, you can do that with just carabiners too. In mountaineering, weight counts a lot.