Japanese swords have certain stylistic commonalities that distinguish them from swords of other cultures, but within this style there were many sub-forms with substantial differences in construction and methods of use. But speaking very broadly, they were versatile cut-and-thrust weapons, practical in a range of use cases.

Curvatures varied, but were never particularly extreme compared to, say, south Asian and middle eastern swords like scimitars and tulwars, or even 17th Century European sabres. Even strongly curved samurai swords of the koshi-zori type tended to force much of the curvature into a relatively small section of the blade near the hilt, which kept the striking part of the blade relatively straight. So generally speaking their curves were not excessive, and well within ranges that would have been considered normal and practical across a wide range of cultures.

The moderate curvature allows for better thrusting performance than a highly-curved pure slashing sword, and the points of samurai swords show that they were very much designed for thrusting. The most common point styles are chisel-like in their shape, with very sharp points, strong tempering, and backed by a thick blade that could support powerful thrusts without breaking off or causing the blade to become bent. Here is a an example.

Compare that to, say, a Burmese dha, another east Asian sword with a similar two-handed slashing design and curvature. The dha was not really designed for thrusting at all. Even the pointed dhas were a bit blunt at the tip, and many were simply squared off so that you wouldn't even think about thrusting. By comparison, Japanese swords were unquestionably intended for decent thrusting performance, and indeed thrusts are very much part of Japanese fencing technique.

Long, curved swords do have an association with cavalry. Thrusting swords tend to get lost on horseback, as the force of a gallop drives the sword deep into its target and it cannot be withdrawn and re-used before it is torn from your hand by the speed of your mount. Slashing swords are easier to keep in the hand, and you don't even have to swing them all that much since the movement of your horse will draw the sword across everyone you pass.

The early distinctive type of samurai sword is the tachi, a long and curved cavalry sword. It was a very successful sword design, and the basic format (single-edged blade, moderate curvature, chisel point, multi-purpose grip) scaled quite well to different sizes, right down to daggers. It was so versatile, it even scaled up to pole-arm format - the naginata, or Japanese halberd - which was definitely used for thrusting.

So, basically, the Japanese just found a general-purpose design principle for long blades that served them very well. The construction techniques for robust long blades in a region without good iron were very complicated, so once they found a system that worked and scaled to different sizes, they had no reason to reinvent it. The relative isolation of Japan meant that competing sword designs from their neighbours never had much chance to catch on, and the skill of Japanese sword smiths ensured that they didn't have much motivation to seek improvements from abroad.

By the Edo period, samurai were essentially unmounted courtiers, far removed from the mounted warriors who had informed the original tachi designs. The long sword was now a daito, much shorter than the old tachi, but still recognizably of the same heritage, and unlikely to change due to cultural isolation and the absence of major military conflicts or revolutions.

But when Japan finally opened up in 1868, and a deluge of new ideas, technologies, and military fashions swept into the country, their swords were curiously resilient. The military swords of choice in the west were neo-classical small swords and sabres. Although the former were quite different to samurai swords, the latter were surprisingly convergent - single edge, gentle curve, versatile cut-and-thrust usage. The similarities were strong enough that there was a brief fashion for retrofitting western-style neoclassical sabre fittings onto Japanese sword blades, before nationalist fashions took over and returned to a more classic Japanese look for mass-produced gunto weapons.

Europe had curved swords from ancient times through the Medieval period and into modern times. For much of this time, the curved swords weren't specifically cavalry swords, and many - especially the shorter ones - were infantry swords or civilian swords for use on foot. In about the 15th century, curved cavalry swords came into common use in Eastern Europe, and slowly made their way across to Western Europe. In the 17th century, curved infantry hangers were in common use in Western Europe, and perhaps somewhat later, curved cavalry swords. The curved cavalry swords coexisted with straight cavalry swords. Some of the straight cavalry swords were intended more for thrusting, but some were designed mostly for cutting (e.g., the British 1796 heavy cavalry sword, and the various pallasches that inspired it).

In the British Army, we see spear-pointed gently-curved cut-thrust swords becoming the standard cavalry sword. The curvature of these blades, as measured in the Japanese way, is about 15mm. Noting that these swords were designed as cut-thrust swords, and that many antique katana have a similar curvature, the curvature of a katana should allow its use as a cut-thrust sword. The other ingredients to make a sword useful for thrusting are a sufficiently pointy tip (which the katana has), and a stiff enough blade, including the tip (which the katana has). Thus, the typical katana is built like a cut-thrust sword.

Unsurprisingly, we see thrusting techniques in Japanese koryu sword arts. Thrusting (sometimes half-swording) are used to aim at gaps in armour in armoured fighting (similar to the use of sword thrusts in Medieval European armoured fighting).

What we don't see in Japan is very long katana to maximise reach, like a rapier. There were longer swords in Japan, and other longer weapons. If you wanted better reach, you would use a spear, or naginata, or odachi. Or a bow or gun.

The other thing we don't see in Japan is an evolution to ultralight swords like the European smallsword. The smallsword, and duelling epee which descended from it, are well-suited to duels to first blood or otherwise to satisfaction short of death. Thrusts into the skull or the chest or abdominal cavity can be deadly, but the smallsword does lack immediate stopping power compared to heavier cutting swords and broader-bladed thrusting swords. There were differences between Japanese and European duelling traditions.

There was a shift in European swords from cutting and cut-thrust to purely or mostly thrusting. The smallsword is largely to blame. The smallsword was the thing carried by the upper classes, and the weapon taught to civilians. Thus, it dominated the teaching of swordsmanship outside the military - smallsword fencing the source of modern foil and epee fencing. Civilian fencing fed back into military sword design. This was a major reason for early 20th century British and US swords being thrust-oriented (e.g., the British 1908 cavalry sword and the US Patton). By this time, the magazine-fed rifle, machine-gun, and modern artillery dominated the battlefield. Cavalry still proved useful in WW1, but little details like whether they used a cut-thrust sword or a mostly-thrusting sword or a purely thrusting sword didn't matter much. Indeed, one of the spectacularly successful cavalry charges during WW1, that by the Australian 4th Light Horse Brigade at Beersheba, was made with bayonets in hand (since the Light Horse were officially a variety of mounted infantry, intended to fight on foot rather than mounted, and thus not equipped with swords). Despite the use of such inferior swords, the charge worked (and the Australian Light Horse started to be upgraded officially to cavalry, including being issued with swords).

Thus, the 19th and 20th century European shift towards thrusting swords in the military was largely irrelevant to combat. Similarly, the Japanese shift from the katana to European-style swords in the late 19th century, and then their shift back to Japanese-Western hybrids and the katana-like shin-gunto was also largely irrelevant to combat. When swords were more important as military weapons, most were cutting-oriented or cut-thrust weapons. There were some specialised thrusting-only military swords (like the koncerz), but most were cutting swords or cut-thrust. Since the katana is a cut-thrust sword (and gives up some cutting effectiveness for that thrusting capability), Japanese swords also followed this pattern.