Mechanically, the bokken is a lever and your hands are the fulcrum point; that means what happens at one end is reflected at the other. In my experience when the tip of the bokken is moving inconsistently with the cut, it is because I am moving my hands inconsistently with the cut... I try to focus on my right hand acting as a fulcrum point and my left hand acting as the motivation of the cut.
As for the impact on your training, I believe weapons work is instrumental in illustrating proper technique, which directly translates to proper taijitsu - distance, timing, footwork, body position, etc.
As for your martial education, I believe weapons work also improves your wisdom and knowledge about the greater history and strategy of combat and combat system development. With greater about about the history of technique, you are more capable to discover more intimate knowledge about that technique.
Ditto the latter points.. But "lever?" I really try to get folks away from lever thinking when it comes to this stuff. Not that it is exactly wrong as stated, but I find it leads some folks in the wrong direction. When cutting what we really want is the center of action to move out along the length of the blade, like the center of action moves from my hara to my spine to my shoulders to my elbows to my wrists it should move likewise from the tsuka smoothly on out into the mono-uchi or kissaki, and then the into the target, ultimately.
It is a shifting center of rotation in a rigid object -- like the "rubber pencil" optical illusion. If the center of action is at the tsuba the kissaki will wobble basically unloaded and be very vulnerable to being shifted by very slight loads -- it is the free end of a lever and longer than the lever in your hands. -- Conversely, if the center of action is moving out the blade into the point of contact, the whole system is dynamically stabilized at contact. But of course if one "forces" the cut it is just applying a reversed leverage to it as that of holding it out -- and likewise defeating the purpose. Many people have this latter problem in cutting.
I try to have them imagine a cut like a weight on a string -- no leverage possible. In extension, I imagine holding the blade while some mischieveous kami is pulling the tip of the blade into my target, and holding that shape
. It is more "tree-limb" like than static lever balance -- like the branch reaches out for more sun. In tai jutsu I try to analogize it to reach vs. grasp -- we cut by reaching not by grabbing -- and reach always exceeds grasp, as we know .
I think the more one practices to generate and modulate furitama and feel the resonance in things like tekubi furi and that same "live" vibe in performing technique, the more control you will have over that expression of what doctors call "essential tremor" which is physiological in nature, ( ~10 Hz) and can be exploited as well as controlled.
We did a drill (in an iaijtusu class) where we stood in extended seigan and maintained connection with the opponent's mono-uchi, close to the yokote -- for five minutes straight. Most of the folks there did find they had or developed a bobble or tremor knocking the other's sword periodically at the end, and this was an indication of insufficient and/or improper form of extension -- which was the point of the drill.
My partner (the class instructor) and I were perfectly still in that way, but we just kinda ended up creeping like mollasses ever so slightly deeper into each others respective sphere's of extension, until ultimately we had shifted from connecting at the front of the mono-uchi to connecting at the back, and each of us was just itching to cut first.