Twenty years ago, during an ice climbing course in the Rockies, an ACMG mountain guide, said to me almost in passing that "we no longer teach boot axe belays." What is surprising is, that two decades later, people are still using boot axe belays.
There are a number of problems with boot axe belays, not the least of which is the uncomfortable position required to instigate and manage the belay. You'll find yourself bent over in an awkward position with your climbing pack riding up around your neck threatening to cut off the blood flow to your brain. If you do manage to maintain this back-breaking position, you'll have difficulty managing the rope. Remember, that a boot axe belay requires you to keep one hand on the head of the axe lest the axe pop out of the snow and your entire belay fail. That leaves only one hand for managing the belay, almost a near impossibility. You may be able to let rope slowly slide through your hand, but taking in rope while maintaining one hand on the axe head and keeping one hand on the rope as a brake is virtually impossible.
Snow anchors are as dodgy as a Harper majority, and the boot axe belay is dodgier than most as the rope running around the axe acts as a lever to pop the axe out. The force on this dubious anchor is doubled by the pulley effect, something you absolutely don't want to do with any snow anchor. Finally, if something goes wrong and you need to get hands free, this operation also is so difficult as to approach impossibility.
So, next time you are tempted to throw in a quick boot axe belay, don't.
No place for a boot axe belay