Martial arts can be a good training to ensure your personal security, if you assume the worst about your tools and environment. If you expect to find yourself unarmed in a dark alley, or fighting hand to hand in a war, it makes sense. But most people do a lot better at ensuring their personal security by coordinating to live in peaceful societies and neighborhoods; they pay someone else to learn martial arts. Similarly, while "survivalists" plan and train to stay warm, dry, and fed given worst case assumptions about the world around them, most people achieve these goals by participating in a modern economy.
The martial arts metaphor for rationality training seems popular at this website, and most discussions here about how to believe the truth seem to assume an environmental worst case: how to figure out everything for yourself given fixed info and assuming the worst about other folks. In this context, a good rationality test is a publicly-visible personal test, applied to your personal beliefs when you are isolated from others' assistance and info.
I'm much more interested in how we can can join together to believe truth, and it actually seems easier to design institutions which achieve this end than to design institutions to test individual isolated general tendencies to discern truth. For example, with subsidized prediction markets, we can each specialize on the topics where we contribute best, relying on market consensus on all other topics. We don't each need to train to identify and fix each possible kind of bias; each bias can instead have specialists who look for where that bias appears and then correct it.
Perhaps martial-art-style rationality makes sense for isolated survivalist Einsteins forced by humanity's vast stunning cluelessness to single-handedly block the coming robot rampage. But for those of us who respect the opinions of enough others to want to work with them to find truth, it makes more sense to design and field institutions which give each person better incentives to update a common consensus.
I would guess that martial arts are so frequently used as a metaphor for things like rationality because their value is in the meta-skills learned by becoming good at them. Someone who becomes a competent martial artist in the modern world is:
Patient enough to practice things they're not good at. Many techniques in effective martial arts require some counter-intuitive use of body mechanics that takes non-trivial practice to get down, and involve a lot of failure before you achieve success. This is also true of a variety of other tasks.
Possessing the fine balance of humility and confidence required to learn skills from other people. Generally if you're going to get anywhere in martial arts, you're not going to derive it from first principles. This is true of most human knowledge domains. Learning to be a student or apprentice is valuable, as is learning to respect the opinions of others when they demonstrate their competence.
Practiced in remaining calm and thinking strategically under pressure. If one is taught to competently handle a high-stress situation such as a physical fight, one can make decisions quickly and confidently even when stressed. This skill is useful for reasons I hope I don't have to go into depth on.
Able to engage mirror neurons to understand and reason about the nonverbal behavior of other humans, and somewhat understand their intentions and strategies. This is useful in a fight and taught by many martial arts, but extremely useful in other contexts, not the least of which being negotiation with semi-cooperative individuals.
Probably pretty physically fit. It's a decent whole-body exercise regimen, and there are numerous benefits to exercising frequently and keeping in good shape. It is probably not the most efficient exercise regimen out there by a long shot, but it may be one that is intrinsically fun to do for a lot of people, and thus it's likely that they'll stick with it.
Almost incidentally, reasonably capable of defending oneself in one of the few instances where civilized behavior temporarily breaks down (An argument with a seemingly reasonable person who quickly becomes unreasonable, perhaps alcohol is involved? I don't know. Fights are low-stakes and uncommon these days but they still happen). This is kind of a weird edge case in a modern society but might non-trivially prevent injury or gain you status when it comes up.
Note that there are a lot of vectors by which one can gain these meta-skills. While there are a bunch of martial arts enthusiasts out there who would probably claim that martial arts have the exclusive ability to grant you one or more of these, I really doubt that's the case. However, martial arts get a pretty good amount of coverage in real and fictional cultural reference frames that we can be reasonably confident most people are familiar with, and it's not a bad example of a holistic activity that can hone a lot of these meta-skills.
It's also worth noting that while the skills involved in interacting with a society of people you trust and want to work with are often different from the skills involved in becoming a competent individual, many of the latter can be helpful in the former. I would much rather be on a team with a bunch of people who understand the meta-skill of staying calm under pressure, or the meta-skill of making their beliefs pay rent, than be on a team with a bunch of people who don't. Aggregated individual prowess isn't the only factor for group success, and it may not even be the most important one, but it certainly doesn't hurt.