It seems to me that we've got a number of challenges to poorly-framed strawman questions. There are a lot of assertions made about martial arts practice developing some thing(s) other than martial skills. You've got to get specific about the assertion before you can offer a valid challenge to it, and I think this conversation has largely failed to do that.
I'm going to leave aside the discussion about what happiness is, and what a "better person" is, and whether a happy person is a better person...you can have that. Instead, I'd like to break the generalized "self-improvement" wah about martial arts into two different assertions (with "self-improvement" being my stand-in term for All That Stuff):
1. Martial arts training uniquely provides an environment for self-improvement (i.e., there's nothing as good).
2. Martial arts reliably results in self-improvement (i.e., if you train in martial arts, you'll self-improve, whatever that means).
I disagree with both of these, with a big "...and yet". I don't think that martial arts uniquely provides an environment for self-improvement. Demonstrably, people improve their physical well-being, their mental acuity, their social engagement and their emotional health through many pastimes and practices, from something as simple as participating in a square dance group to joining a spiritual community. Furthermore, I don't think there's any single "self-improvement" goal for which there isn't a better and more direct alternative to martial arts. Want to improve your physical conditioning? A focused and dedicated exercise program works better than martial arts training. Want to deal with anger or fear or other destructive emotions? Therapy or some spiritual practices are the way to go.
I don't think that martial arts reliably provides an environment for self-improvement. How could it, when most people don't even stay long enough to develop competence at the simplest physical technique, let alone make some kind of mental/emotional/spiritual breakthrough? And don't we all know people who have been training for years and yet who still seem "stuck" as people?
And yet, I do believe that budo training provides a very rich environment in which "self-improvement" can take place. It won't happen for everybody, because not everybody is ready for change. But for those who are, budo training has many aspects and experiences that serve as powerful catalysts. The opportunity is there. And it's grounded in the physical, which I think is a helpful and accessible way for us humans to learn. Back in my coaching training days, they told us to "activate prior knowledge": find something that the student already knew, and relate to that. We humans begin as physical beings, and I think we always retain our ability to relate to the physical, even if other ways of understanding elude us.
So, that's my take on it. No, training in martial arts doesn't "lead to self-improvement". No, training in martial arts isn't the only (or even the best way) to "self-improvement". No guarantees that it will happen to you. But budo training is rich in opportunities for "self-improvement". If you train for any length of time and fail to pick up on any of these opportunities, either you took advantage of other opportunities elsewhere...or it just isn't your time to change that way.