Oh, but you do.
No upward force and the pole hits the ground. If the impact was near a cliff or in outerspace, then you could have a full caber-like rotation (though technically that's not the best description since throwing a caber implies a vertical force).
No.
If hit from the side, and near the top, like we expect, the pole will rotate around its center of mass after the base is broken. Since this impact happens very, very fast, this will leave the pole momentarily in free space. There is no vertical force required to get it completely airborne.
Having said that, it is also possible for the collision to be significantly inelastic. At impact, the upper part of the pole may deform and actually catch on the aircraft. From there it's anyone's guess as to where the pole goes -- whether it merely gets swept along before buffeting or rebounding free, or if the deformation means the leading edge actually lifts it up like a dustpan under a chunk of FOD.
In a caber toss, there is really very little vertical force applied anyway. What happens instead is the throw is done after the caber starts to lean. This creates a force couple, with the (vertical) force at one end and gravity pulling down in the middle. This rotates the caber, but rarely lifts it much. One gets good caber distance by timing the rotation properly to get it to topple in the right direction after it hits the ground. One does not throw it upward very far, unless one is absurdly strong...
Please stop trying to lecture me about physics. You have no idea what you are talking about, and you are embarrassing yourself. This will not help you learn in the slightest.