Over the last 200 or so years, three things have driven modern European martial arts towards simplicity:
Duelling. Formal duelling with rules prescribing equal weapons, allowing duels to be stopped with honour satisfied after first blood, and potential legal problems if a death occurs have enabled and encouraged a shift towards fast light weapons with limited stopping power. As a result, the smallsword - fast, light, and thrusting-only - became the premier duelling sword in much of Europe. This, in turn, influence European sword-based martial arts, with the foil being in essence a training smallsword. As the smallsword disappeared as a dress item, the duelling sword could lose much of its ornamentation, and become even more specialised: this was the duelling epee (see https://fencingclassics.wordpress.com/2011/12/21/weapons-eight-dueling-epees/ for some discussion and examples); the fencing epee is the training equivalent. As purely thrusting weapons, offensive techniques don't include cuts, and defensive techniques can be fewer since there is no need to defend against cuts. Since like weapon will be used against like, techniques for defending against other weapons aren't needed. With grappling, throws, disarms etc. considered inappropriate, these need not be taught either. In contrast, earlier European martial arts systems included sword vs other weapons, grappling, punching and kicking while using a sword, disarms, throws, etc.
Military training. As the sword retreated from the battlefield as a serious infantry weapon, it remained a major cavalry weapon well into the 20th century. While cavalry also used pistols/revolvers and carbines, the sword was still considered useful and important, and cavalry trained in its use. Military training in swordsmanship typically focussed on a simple and basic set of techniques, suitable for drilling the soldiers in a unit en masse.
Increasing restrictive rules for combat sports. This is particularly noticeable for boxing. Up to the early 18th century, there were few rules in boxing, and a wide range of techniques no longer allowed, including grappling, throws, and kicks, were allowed. Jack Broughton's rules of 1743 (see https://babel.hathitrust.org/cgi/pt?id=njp.32101072325507&view=1up&seq=75 pp 51-52), a first major step towards modern boxing rules, were not very restrictive. They prohibited hitting a downed opponent, and prohibited holds below the waist when wrestling (no grabbing thighs or pants). The London Prize Ring Rules of 1853 were more restrictive, adding to Broughton's light restrictions bans on headbutts, low blows, gouging and biting. The Queensbury Rules of 1867 further banned wrestling. Further rules as modern boxing evolved banned kicking, kneeing, elbows, and a variety of hand techniques such as hammer fists and backfists.
In turn, many Asian martial arts have been driven towards elaborate techniques by an orientation towards performance. Some Chinese martial artists worked as street performers, and spectacular techniques are more likely to free the spectators from their money than simple straightforward techniques. In addition, flashy techniques were used as a recruiting tool (this is one reason why Taekwondo developed so many jumping and spinning techniques).
With Asian martial arts (and especially unarmed martial arts) largely divorced from military training, and often free from being tested in action, styles could develop a wide range of elaborate (even if often ineffective) techniques.
There were pushes towards simplicity in Asian martial arts. This can be seen in the development of Judo from jujitsu and Japanese karate from Okinawan karate. In both cases, teaching techniques were developed for mass instruction in physical education in schools, with simplification of techniques, and there was further simplification in the interests of safety. The set of techniques used in kendo is small compared to the range of techniques seen in old schools of Japanese swordsmanship.
Indeed, the example of kendo shows that if we compare like martial arts with like, we see little or no difference in the elaboration of techniques. (European fencing shows more elaborate techniques than kendo, with behind-the-back techniques (see 6th photo in http://www.socaldivision.org/the-socal-musketeers-ryc-epee-rjc/ for an example), jumping techniques (example and example), and more.) If we compare performance-oriented Asian martial arts with European stage-fighting, we find similarly flashy and elaborate (and often ineffective) techniques in both. In European wrestling styles that were less restricted by rules (such as catch wresting ("catch-as-catch-can")) including a wide range of submission techniques comparable to techniques in jujitsu and derivatives. Going back in time to Medieval European martial arts, centred on a range of weapons used against a range of weapons, including punching, kicking, and wrestling with weapons, and the possibility of multiple opponents, we see a similar range of techniques (and often very similar techniques) to those in Asian multiple-weapon martial arts (see https://wiktenauer.com/ for a range of fightbooks showing these techniques).
There are still surviving European martial arts of considerable complexity. These are less widely known than boxing, wrestling, and fencing, but they have avoided much of the simplification seen in arts such as boxing. Two examples are:
Jogo do Pau: https://www.youtube.com/watch?v=ajMnwhKgr1s
Finally, while many martial arts - such as boxing, many styles of wrestling, and kendo - only allow a limited range of techniques, a great deal of skill is needed for the optimal application of those limited techniques. A limited set of techniques does not necessarily make a martial art easy to master.
Ehh... They don't.
Modern combat sports have very streamlined techniques based on what works in that particular rule-set at that particular time.
Traditional european martial arts contain a variety of elaborate techniques depending on the time period, style, and area that you are talking about.
There is also a large amount of overlap between Asian and European martial arts techniques from a historical perspective.
One thing that makes European martial arts SEEM more straightforward is the fact that when taught in one's native language they tend to include explanations for techniques that avoid any mysticism or other nonsensical ideas.
BUT, if you were to study, for example, Judo, Kendo, or K-1 style Karate, you would find that they are very straightforward in their explanations and techniques especially when given in their native language.
A combination of language barriers and charlatans wanting to promote nonsense has lead to the perception that Asian martial arts contain a lot more "elaborate techniques" than they really do. And the promotion of modern combat sports has led to the idea that western martial arts do not contain more subtle and complex movements that are perhaps less suited to a sporting environment.
Neither of these things is true.
A brief look at the historical records of martial arts from 2 countries will give us an idea of what we are looking at here:
Italy and Japan
I chose these countries because they both have robust martial traditions that span centuries and lead in to modern combat sports, but are, definitively examples of Asian and European countries.
And to zoom in a little further, though you could easily generalise, we will look at 2 martial arts "schools". One from each country.
From Japan, the Tenshin Shōden Katori Shinto Ryu, which just means "The True and Correct school of the God Katori"
From Italy, Fiore di Liberi's "Il Fiore di Battaglia". Which just means The Flower of Battle. And what we are going to look at to answer your specific question is the curriculum of each school.
First, the Japanese school. The primary curriculum includes Sword, Staff, Glaive, Spear, Throwing Weapons, and Unarmed fighting. Ōtake, Risuke (2007). Katori Shinto-ryu: Warrior Tradition. Koryu Books.
Second, the Italian school.
The Italian school primarily includes, Unarmed fighting, Baton, Sword, Poleaxe, Dagger, Spear and Mounted combat. Fiore de'i Liberi. Fior di Battaglia [manuscript]. MS Ludwig XV 13
You can see that the curriculums have more overlap and similarities than they do differences.
Another thing that tends to make the Asian martial arts seem more elaborate is the fact that some Asian schools have a continuous and unbroken tradition going back centuries. Most European martial arts have been lost, reformated, rediscovered, and in general have more breaks than a jigsaw puzzle in their transmission.
This means that you don't have as many people that have been practicing Historical European martial arts for 30-40 years as you do Asian martial arts. 30-40 years of dedicated practice can make anything look elaborate, elegant, precise and amazing to watch.
I have limited this answer to reviewing those schools of martial arts that are more complete in their approach, but a similar approach can be taken to looking at any comparable martial arts from Asia to one from Europe. Jiujitsu, Judo, much of Kung Fu, Etc, are all forms of Wrestling. Europe contained so many forms of wrestling at one point that it just becomes easier to group them all in to Wrestling and call it good. At one point, moving from one county in England to another was enough to change the style and type of wrestling taught there.
