Forest trees in good form, single stemmed and tall, there's definite and certain mechanical advantage to using the Humboldt:
1. Falling the tree uphill, absent of any crown, the butt of the tree will rest back on the stump. In this case the Humboldt helps keep the tree from shooting down the hill. The negetive slope of the diagonal, in effect, extends stump shot. In order for it to work out that way, at times, you may have to purposely cut the stump high to catch and hold the tree.
For the climber, the Humboldt cut, by the same virtue, can help keep the butt of a top from firing back over the cut,,,, if it were to hit another tree and stick or hold momentarily. If a top does that it will exert back pressure on the cut!! Something you really don't want to have happen, no matter. It's just plain bad practice to go ramming your tops into or through other trees period. Same goes for the faller.
2. The Humboldt cut applied to sidehill or downhill falling, the negetive slope can promote the butt of the tree to slip off the stump and reach the ground before the top. The tree will lay from the butt out to the top. Saving a particular valuable tree from shattering in a swale. There's some fine points that should be elaborated on, but it gets real in depth, and not all that necessary to know just to understand the underlying principal. The down-slope of the cut, like a hillside, promots things to slide, slip, fall off or roll down.
The particular tree or situation can void the mechanical effectiveness of the Humboldt and render your hopes of it working totally in vain.
Whether the Humboldt is really needed at certain times is highly debatable. Most fallers I know use the cut only as a matter of convienence. It's just what they are use to using. It's easy for them.
On the bad side,,, the Humboldt method leaves a higher stump and for that it's somewhat wasteful when you stop and look at a whole layout of stumps that would be lower if the conventional cut was used instead.
I'll add to that, while the issue of waste is a valid point, at the same time the timber compaines and log buyers don't appreciate having the diagonal or snipe coming in on the butt logs. They prefer all logs to be square on both ends. So while the Humboldt may keep the log buyer happy, the forester is out citing fallers for making high stumps at the same time. It's an issue, I believe, will never be resolved.
If you don't have any use for using the Humboldt why use it? Believe me, for the far majority of trees the conventional cut is all you'll ever need. If you get into tall single stemmed forest trees that in good form, then you'll find more reason and opportunity to use the Humboldt.
And gunning the undercut? Burnam says it right.