A similar variety of martial arts developed in Japan as in any other country, but more survived in Japan because it never had a military revolution as extreme as the one that took place in Europe until the emergence of the modern martial arts market.

Europe once did have a significant martial arts tradition, today called Historical European Martial Arts (HEMA). Many of these arts were similar in practice and technique to traditional Jiu Jitsu and Kendo. HEMA declined with changing military technology. Between 1400 and 1700 AD, significant changes to battlefield tactics meant hand to hand, single-combat scenarios were increasingly rare. Some of the developments during this period include the proliferation of gunpowder weapons and the equally significant resurrection of the Phalanx in the form of the Tercio. Armies re-directed their attention from training individual combat techniques to drilling formation tactics, reloading, and marskmanship.

In contrast, Japan's military revolution was less severe during the same period. Though arquebuises were eagerly introduced in Japan, the Sengoku Jidai ended shortly after their introduction. The post-unification Japanese government enshrined the privileges of the Samurai class, who maintained the traditions of Jiu Jitsu long after they were useful on the battlefield.

After the Bakumatsu (fall of the Shogunate) in the 1860s, former samurai passed down the techniques of traditional Jiu Jitsu, which were later codified into Judo. The reason Judo didn't fall into obscurity like HEMA is that this was around the time of the industrial revolution, which caused a significant uptick in disposable income, human cross-country mobility, and urbanization. In other words, Jiu Jitsu survived long enough to meet the modern martial arts mass-market. Had Japan continued its internal conflict into the 18th or 19th century, it is very possible that Judo today would be as obscure as most historical European martial arts.

This, of course, only addresses a third of your question, as you had also asked about Karate and Sumo. Sumo was a competitive martial art akin to freestyle wrestling or fencing, a highly "gamified" martial art with many rules and which wasn't full contact. In this respect, Japan is not so different from Europe in having such a competition.

Karate, however, was another versatile martial art which developed in Japan, but it followed a completely different course than Judo. While Judo was a codification of samurai grappling techniques, Karate was derived from Southern Chinese martial arts, most notably White Crane Style, by the Okinawans, who had significant contact with the Chinese. Like Judo, Karate had come at the right time to benefit from mass-market promotional opportunities that industrialization offered.

This all begs the question of how Chinese martial arts survived long enough for some Southern arts to be codified into karate in the first place. The situation in China was in this respect similar to the one in Japan as China had far fewer security threats than most European powers during the same period, and, because of this, also experienced a far lighter military transformation than what had occurred in Europe. Average Chinese taxation rates as a percent of GDP ranged between 6 and 10 percent during the 15th century to 18th century period, while in some European countries, they exceeded 50% - indicating a far higher rate of military spending per capita in Europe. Hand to hand combat with blades, polearms, and spears was still a major component of Chinese battles well into the 19th century.

So in summary, it wasn't that East Asia developed more martial arts, but simply more survived until industrialization allowed them to transition to a mass-market, hobby/sporting business model. Centuries of perpetual war in Europe had brought major military changes which were slower to develop in the comparatively more peaceful countries of East Asia.

Sources:

Malliaropoulos, Nikos et al. Judo, the gentle way.

Rosenthal, Jean Laurent. Fiscal Regimes and Economic Growth; China and Europe.

Broadberry, Stephen et al. China, Europe and the Great Divergence: A Study in Historical National Accounting, 980-1580.

Jaquet, Daniel et al. Historical European Martial Art a crossroad between academic research, martial heritage re-creation and martial sport practices.

Johnston, Ryan. Contemporary Martial Arts: Self Expression or Self Oppression?

u/Interpine has already discussed the survival of Japanese martial arts into the modern day. Still, despite Japanese military technology having sat in the pike-and-musket era from the 16th century through to the mid 19th century, many, if not most, martial arts schools (ryu) disappeared after the end of the Edo Period. Also, other regions - notably China - also "benefited" from this survival of pike-and-musket to the mid 19th century (or later).

Two keys to the survival of some of the Japanese martial arts were:

1. Their transformation into sports.

2. Their use in physical education in schools to instill martial virtues (and physical fitness) in the school system.

These are not uniquely Japanese - the prominent surviving Western martial arts (boxing, wrestling, fencing, archery (and some equestrian sports, if you wish to add them, and rifle and pistol shooting)) also benefited from the same.

I would question the premise of Japan being such a fertile ground for the development of martial arts. With a broad-brush picture of martial arts, there is little to say that Japan in prominent in terms of number. Considering martial arts that survived into the 19th century, we have:

.	Japan	China	Korea	India	Europe
sword	yes	yes	some	yes	yes
spear	yes	yes	some	yes	lance
archery	yes	yes	yes	yes	yes
gun	yes	yes	yes	yes	yes
weapons while mounted	yes	yes	yes	yes	yes
wrestling	yes	yes	yes	yes	yes
unarmed striking	some	yes	some	yes	yes

At this level, Japan doesn't appear at all exceptional. If we zoom into the level of individual schools and styles, we see a proliferation in Japan (and also in China). In Japan, this appears to be due to the formal naming of styles and the passing of authority from headmaster to headmaster (soke) in succession, sometimes with parallel head instructors (shihan). In Europe, there were many fencing schools, but they were less likely to descend from instructor to instructor as a codified style/school. If you were to count individual European fencing schools, you might find little difference in numbers between them and Japanese sword ryu. If we broad-brush European fencing schools to the level of "French", "Spanish", "Italian", then the Japanese schools outnumber them.

For unarmed martial arts, the fertility of Japan depends very much on how broad our brush is, and when we look at the scene. If we look at the 19th century, we see many local styles of wrestling, many of which became extinct or nearly so, as a narrow range of wrestling styles came to dominate national and international sport. In Japan, these styles of wrestling are jujutsu (often including some striking), and in Europe these are various styles of "folk wrestling" (often including some striking).

In the West, we have boxing as a martial art/sport with ancient roots. In Japan, we have unarmed striking as a minor component in jujutsu (largely removed as modern arts like judo and aikido grew from jujutsu). While top-level boxers might well become celebrities, boxing was, compared to martial arts like fencing and equestrian arts, rather lower class. The prominent Japanese martial arts being largely the martial arts of the ruling elite, it is unsurprising to see a lack of striking-focussed unarmed martial arts in Japan. This gap is perhaps what enable karate to be developed into a Japanese martial art from its Okinawan roots (karate only became a Japanese martial art in the early 20th century).

In summary,

• Japan developed a similar range of martial arts to other regions, including Europe.

• The formalisation of styles appears to have led to a proliferation of formal schools/styles that had an existence beyond their founder's school/dojo/salle.

• The use of martial arts in physical education in schools led to the growth of kendo, judo, karate, and atarashii naginata (sport naginata), often at the expense of older traditional martial arts. Especially if we discount boxing, wrestling, and fencing in the West as "martial arts", this makes Japan prominent in martial arts.