But when the endstate is "becoming a better person" (which IMO is both undefinable and a copout for lacking martial skill), how one measure that?
If "better" in undefinable, then there is no way to measure it, so the question doesn't really make any sense. You can't measure something that you refuse to define.
Moving beyond that, though, coming from the perspective of a budding researcher in the social sciences, here are some different components I think about for measuring this.
1) First you have to define what you mean by "a better person" in a way that could be measured. Do you mean that they feel they have more self-control? Behave more ethically? Able to remain more calm when the feel stressed? Of course, you may mean all of these things, so you might end up having to do many, many research projects if you want to capture many, many different characteristics. Another issue would be determining whether you want to focus on self-perceived "betterness" or some outward expression (change in actual behaviors) of "betterness."
2) Once you know what you want to measure, then you find some kind of scale that measures it, or build your own scale. There are already tons of personality scales, scales of perceived self-efficacy, scales of mental affect, systems for rating both self-behaviors and behaviors of others, and tons of other stuff, that have years of research behind them.
Or if you want to, you can design your own scale and do some rigorous testing to make sure it is coming close to measuring what you want it to measure.
3) You need a research design. Probably something pre and post. Your participants (hopefully you could start with a few hundred) take your scale, get a score, then take aikido lessons for a year, they take the scale again. If you have enough, you might divide them into two or three groups, each pursuing a different activity so you can compare them. Perhaps Group A does aikido, Group B does MMA, Group C does dance. It would also be great ot have a control group that does nothing.
4) Compare the pre and post results to see if there was any change, then compare the amount of change between groups to see if any activity stood out as better than any of the others.
So, in the end, if one defines "better" as an undefinable characteristic, then yes, attempting to measure it is silly. If there is a way to point to more specific pieces of "betterness", then it becomes easy to think about measuring change.