Honestly, in all the discussions we've had on this forum over the years, I have yet to find a definition of "TMA" which reliably differentiates al the arts which are widely regarded as TMA vs those which are not. So I'm not going to worry about whether BJJ does or does not fall into that category. But I can do a compare and contrast regarding the historical development of BJJ vs Karate/CMA, and you can do with that what you will.
CMA
Discussion of the development of CMA in general is, I think, such a broad topic as to be almost pointless. Especially if you're going back centuries, which means you're pre-dating the modern conception of distinct named styles/lineages. But I'll hit some broad strokes.
You had military fighting systems, primarily bow, spear, and sword. Just as in the West, these have mostly been lost or transformed beyond recognition from their battlefield forms. I can't remember seeing a single CMA sword form which resembles the use of a sword when fighting in formation.
You had folk wrestling systems, which could be used in sport, self-defense, or in military training to build fighting spirit or as a supplement to close-range fighting in armor.
You had a wide variety of systems for armed and unarmed individual fighting. Probably these were originally the informally gathered experience of an individual family or village, but over time economic and cultural pressures led them to become "branded", if you will, with distinct names and claimed lineages and so on. This branded stylification could serve a number of purposes.
The school could provide a social group identity for students.
Defining a style as fitting within certain parameters could provide a teacher with a way to differentiate his school from competitors.
A teacher who could establish himself as the authority within a certain style could supplement his income and social status by controlling the flow of information within the school. "Secret" techniques or principles could be reserved for those who showed special loyalty or who paid extra.
(Confucian ideals of hierarchy certainly fed into that.)
Some practitioners tried to make a name for themselves via challenge matches. Others went the theatrical route, making money by staging shows for entertainment. (This eventually reached its logical conclusion with the creation of modern wushu by the CCP.)
Karate
Anyway, some branch or branches of an undetermined Hakka CMA reached Okinawa, possibly blending with the indigenous "Te" grappling art. In Okinawa, it was practiced as an unarmed fighting/self-defense art, focused primarily on dealing with untrained opponents at close range, and was trained in a relatively informal way - no special uniforms, ranks, standardized curriculums, or officially distinct style names. Since we don't know the exact CMA origins of this original Okinawan karate, we don't know if any or all of those features were representative of how the art was practiced in China.
With the official annexation of Okinawa by Japan and the efforts of Funakoshi to popularize the art, a rapid process of "Japanification" took place, especially for those branches of Karate which took root in (non-Okinawan) Japan. Uniforms, belt ranks, deeper and wider stances, fighting at longer ranges, more focus on striking vs grappling, standardized curriculums, officially distinct styles, sport competition, etc, etc. All of this had to do with fitting in with the Japanese culture of the time rather than combative requirements.
The Japanized form of Karate took root in Korea, where the native martial arts had been mostly suppressed under Japanese occupation, When that occupation ended after WWII, the Koreans quickly rebranded Karate. They originally kept both the technical base from Shotokan and the cultural trappings (uniforms, belt ranks, standardized curriculums, various formalities, etc). But over a generation or so, they reworked the technical base to be visually distinct from Japanese Karate (more kicks, higher kicks), changed the name of the art, created new forms, changed up the competition rules, and inserted Korean language and symbology in place of Japanese. None of this had anything to do with combative needs. It was strictly to do with resentment of the Japanese occupation and the desire to have a distinctly Korean cultural art. Despite those efforts, you can argue that Tae Kwon Do is technically no more distinct from Shotokan than Shotokan is from Shuri-Te.
BJJ
Japan has a centuries long tradition of various arts that I will call part of the overall "jujutsu" family, mostly involving various forms of grappling, locking, throwing, and pinning. Some date back to battlefield use in armor, but these have almost all been lost over time. Most of the surviving systems date from peacetime and were used for self-defense, law enforcement, or individual fighting.
By the late 19th century, jujutsu had largely fallen out of favor in Japan. The country was in the process of modernizing and Westernizing. Jujutsu was seen as an outdated relic of feudal times, there were few students, and instructors could rarely make a living at the art. Jigaro Kano was one of the few who had an interest in jujutsu. He studied Tenjin Shin'yō-ryū and Kitō-ryū, and within 5 years he had opened his own dojo, teaching his personal conception of a modern reinterpretation of these arts - Judo. This conception was very much his own, but it succeeded largely because it fit the Zeitgeist. It was presented as modern, scientific, and pro-social. Even though the art was called Kano jujutsu for a while, it wasn't long before apostles were preaching that Judo was a modern art entirely distinct from antiquated jujutsu. Kano also introduced some of the cultural elements (belt ranks, uniforms) which later spread to Karate and are sometimes seen as markers of a "traditional" art.
Judo may have a bit (its main reason for being was sport)
Sport was definitely not the original reason for Judo. That came later. Kano saw Judo as a path to self-improvement and the betterment of society.
In the early 20th century, a judoka named Mitsuo Maeda set out from Japan on a world tour to promote Judo via demonstrations and challenge matches. He arrived in Brazil in 1914 providing demonstrations and briefly opening a school. After a couple of years, he resumed his globetrotting fight career before returning to Brazil and teaching Judo mostly to Japanese immigrants. During his career of fighting and public demonstrations, he appears to have mainly used the term jiu-jitsu for his art rather than Judo, and so that was the name by which it initially became known in Brazil.
Among the students at Maeda's original school in Brazil were Carlos Gracie (of the famous family) and Luiz França (much less famous, but with his student Oswaldo Fadda he formed a parallel lineage to the Gracies that today is technically indistinguishable from any of the Gracie lineages). It's uncertain whether Gracie and França learned directly from Maeda or from one of his students, but that doesn't really matter much for the history.
Since the Gracies did much more to popularize the art than the França-Fadda lineage die, I'll simplify the rest of the story by focusing on them.
Carlos Gracie took whatever he learned in his short stint (no more than a year or two) at Maeda's school and taught his brothers. They liked fighting, they liked making money, and they liked being famous. So they decided that a good route to fame and fortune was to promote their art and themselves as teachers by means of public challenge matches.
Normally, such a minimal training background wouldn't be a great start for a martial artist wanting to make a career out of fighting professionally. But the Gracies had a few factors in their favor.
- They trained really hard and sparred all the time
- They had a lot of family members and a lot of students, all of whom trained and sparred hard
- They got lots of real fight experience, not only in professional prize fights, but also in dojo storms and fights on the streets and the beaches
- They cheerfully appropriated techniques and training methods from other arts that they encountered (in particular catch wrestling, freestyle wrestling, Judo, and Sambo)
- In their early years, before they had fully built the skills they became known for, they had no problem with arguing and negotiating the rules of a match to be in their favor.
- They figured out early on that most people have no clue about how to fight on the ground. So they focused their fight tactics on dragging opponents down to the ground for newaza. The more time they spent training in this specialized area of combat, the greater their advantage became once they got to the ground.
All of this was very much not in keeping with Kano's original vision for Judo. But it was a good cultural fit for Brazil, where machismo and braggadocio were valorized.
BJJ evolved fairly quickly over the first 70 years of its existence as a distinct art from Judo. (Some would still call it Basically Just Judo, but I think the difference in tactical approach justifies considering it a distinct art.) This evolution was due to the combined efforts of not only the many members of the Gracie clan, but all their students, and the non-Gracie instructors and competitors and the practitioners of other arts that they trained with and fought against. (Helio Gracie in his later years tried to take credit for single-handedly inventing the art, but it's worth nothing that he tried this only after his brothers had all passed away. In any case, the historical record is clear that he was just one important influence among many.)
After the first UFC, BJJ became popular worldwide and the number of practitioners grew exponentially. While the Gracies are still important members of the BJJ community, dominance of the art has passed out of their hands. (Both in teaching and competition.) With that spread of the art, the evolution sped up even more. The worldwide jiu-jitsu laboratories are constantly at work. The way I learned to do fundamental techniques twenty-five years ago is no longer the best practice. And new techniques and tactics are constantly being invented.
This innovation, BTW, is almost exclusively based on functionality. But there are different contexts for that functionality - sport grappling competition under particular rulesets, self-defense, MMA, law enforcement, etc. Which means that development in one context may not translate to another. One way I narrow down the innovations that I try to keep up with is to look at which principles and techniques are applicable in the largest number of contexts.
So ... that's a lot of blather. I apologize to anyone who is a slow reader but still made it all the way through. But based on all that, is BJJ a traditional martial art? Is TKD? Is Judo? Is Japanese Karate? Is Okinawan Karate? I don't know. Give me a definition of TMA and you tell me what fits and why.