I am a little confused, are you specifically asking why the martial arts systems of the west did not share the same system of schools found in East Asia?

To answer your question as a martial artist, we do have that. We have all kinds of western martial art styles such as: Boxing, the many forms of wrestling, savat, kick boxing, bjj, kajukenpo (spelling?), krav maga, etc.

The key difference is that most western martial arts have strong traceable lineages and emphasis on actually fighting. This means that if someone wants to found a new style, its a significant break in method and approach IE judo and bjj. Because of the live training approach western martial arts are always developing.

Where as eastern martial arts tend to have a lot of nationalism and spiritualism caught up into the art and they tend to focus on kata over actually fighting. There are exceptions to this, mostly in the Japanese arts like judo and kyokushin. But because of the emphasis on kata and aesthetic there is less evolution its much easier for people to start a new style.

Let me throw in my opinion as someone who has done boxing, kung fu, and judo for over 10 years collectively.

For your first query, if it seems like there are much more unarmed combat styles in the Eastern pugilistic world compared to the Western world, that is because it is true. By and far, Eastern martial arts styles in sheer quantity outweighs Western ones. While the Western world also has several ancient martial arts forms, like pankration from Ancient Greece, the sheer amount of ancient martial arts from China and India numbers in the hundreds.

Why is martial arts so prolific in Asia then? Well, I cannot speak for India, but as for China, and kung fu in particular, there is a saying that "All martial arts descend from Shaolin Temple". My teacher in kung fu told me a story once, about how Shaolin was originally a peaceful Buddhist temple in Henan Province within China. However, the repeated onset of war, resulting in Shaolin being attacked, pillaged, and destroyed multiple times, forced the Buddhist monks there - the Shaolin monks - to use their deep understanding of Zen, and their innate devotion and mental fortitude, to come up with martial arts systems for self defense. Every time Shaolin was destroyed, these monks managed to escape using their martial arts expertise, and would blend in with the general population until the conflict and war abated so they could go back and rebuild the Shaolin Temple. While in hiding, those who hid in the countryside and the woods observed the behavior of wild animals and came up with the many different "animal" Shaolin styles. Those who blended into local villages taught the inhabitants these styles as a means of self-defense, and subsequently these styles were passed down through the generations, and preserved throughout the generations even after the Shaolin monks left to rebuild. The Chinese/Eastern tradition of passing down martial arts is also quite lenient about 'branching off' to form your own school, which can then evolve into something very different. Even now, in Eastern martial arts, so long as the master of the school approves of your skills enough to give you a scroll stipulating you have the right to teach, you can open your own school. This applies to nearly all Eastern schools, be it Chinese, Korean, Japanese, etc. Many authentic schools can therefore be asked to produce this scroll, which can trace their lineage back some hundreds, even thousands of years. Over time, as more styles branched off from the original, you had hundreds of styles emerging, hence the sheer number of Eastern styles over Western ones. Many of them have shared roots, and some experts, like my teacher, believe all of them are sourced from Shaolin Temple. This also explains the codification process somewhat.

Why didn't Western styles develop like the Eastern styles? The Western world has a long tradition of martial arts, after all, many sourced from Ancient Greece. One theory is that the development of the art of war was fundamentally different. In China, India, Japan, and much of the Far East, war never grew to revolve around heavy armor, shields, massed gunpowder, etc. - essentially, war in the Far East never got to the point where martial arts was nullified and made into a hobby - this only happened in the modern world. For a long, long period of time, martial arts was very much relevant in the Far East as a weapon of war. So, while unarmed styles prevailed in the Far East, in the West, they didn't "die out" so much as become phased out into hobbies, in favor of weapon-based martial arts styles, from the armed gladiators of Ancient Rome, to the swordsmanship and fencing of Europe over the centuries, to later marksmanship.