The idea of cutting through things is not actually an important part of combative swordsmanship anyway.
Yes and no. The first factor is the school's philosophy of combat. Some have larger cuts, others more slashing cuts. Kind of the lop it off vs. death (or incapacitation) by cuts to important areas. So saying it's not important begs the question of the style in question. Secondly, the reasons for practicing tameshigiri (test cutting) vary tremendously from style to style. Some use the test cutting to use analyze the cuts later to evaluate hasuji, angle, etc. This also gets to Mr. Rubin's comment. People hear things like "don't pull the cut so much" (sometimes told to those who cast the blade too much). They infer from that statement that the blade isn't to be "pulled". That may or may not be true depending on how the style actually performs the cut. Sometimes the very structure of a "proper" cut in a style results in the "natural" arc created by the way the cut is performed to result in a more subtle drawing of the edge across the target. This is of course also facilitated by the shape of the blade and to some extent the shape of something being cut, but the shape to the thing being cut isn't nearly as important.
If you really want to get complicated, talk about arts that originated much longer ago where the cuts are also at times intended to deal with an armored opponent. Now we're talking harder targets and specific ways of cutting to deal with the pieces sewn together hanging down to protect the throat, for instance, vs. a cut that may in fact be more of a percussive strike to deal with a head possibly wearing an iron helmet. Do either of those wrong and you will "fail" in your "goal" of the cut (whatever that might be) and in some cases you dramatically increase the risk of damaging or even breaking the blade.
So again it is rather complicated and it always cries out for a larger context.
The purpose of cutting practice varies rather extensively from style to style. Some do it rarely only to check on form. Others do it vastly more regularly (think the Toyama and Toyama derived groups). So not only are there all sorts of differences in why the cutting is done the way it is done, there are also differences in how they evaluate what would be a "successful" cut, using considerations that often greatly transcend the rather mundane "I cut it" aspect and are usually completely opaque to the casual observer outside the tradition.
And on cutting the head. Take a slightly curved blade and slam it down straight down on a tomato and let me know how well that works. Next either push or draw it through. Try lots of draw, a little, etc. Next try it on different things and you start to understand the interplay between sharpness, moving the edge, downward pressure, and then other factors as well. And how the balance of those things will vary tremendously depending on what you're cutting, why you're cutting it, whether you're trying to score it, cut it halfway, or all the way.
But I'm done on this topic. I've received enough angry emails from those who feel I'm putting down Aikido as a global thing. That is not the case as it seems to vary tremendously in my experience. I also recognize that not all styles are concerned even one little bit on how it would work with a live blades as their considerations are more on aiki, blending, etc. Yup, and that's all good. So basically, *(#$ it all. Y'all carry on with whatever you're doing. Obviously from some of the emails I've managed to generate I've either got to quit putting my identifying info on posts or just quit talking about it. Carry on, you're all beautiful snowflakes and your swordsmanship is fantastic. No one will ever defeat you or your sensei/sifu/grandmaster/professor. Carry on.