One of the biggest problems that I can see for use of ice axes while in the tree is the added strain you're putting on your arms to do things that we normally use our legs for. If I had to haul myself all over a tree using ice axes, my arms would feel like spaghetti while holding the saw.
And you'll notice the name "ice" axe... probably a reason for that. If it was an efficient form then perhaps we'd have "tree axes". But I think it's an interesting idea.
I don't know if I actually mentioned it on the forum but I was looking into using Ice climbing crampons when I first got into trees. Couldnt see why any normal person wouldn't want like 24 actual spikes to use instead of being limited to just two! (just didn't make sense!)
Since then I've noticed that we use gaffs because it's more efficient than having 12 spikes all around each foot. Could you imagine all the rope you could mangle? Ugh... And the spike we do use is inline with the leg, which lowers torsion on the foot.
It may be a bit difficult at first to understand why some things are or aren't used, but after a while it should all kind of make some sort of sense.
Like Ekka says though, hey if you can video it, lets see it.