(DISCLAIMER: the following is just meant to answer the OP’s question in the most unfair, dishonest way and shouldn’t be taken literally. What the previous posters say about Koto swords is true and this shall only be taken as a caustic reply in which most, if not all is exaggerated!)
Soooooo! What makes Koto swords so desirable?
Let me be, for a moment, the fly in the ointment!
1) TRADITION: just as everything was “better before”, so it is with swords. Japanese swords were made by a very rigid society that engendered a rigid hobby. So obviously, past and venerable techniques are necessarily better than newer ones, even if, at the time, they were only dictated by what swordsmiths had at their disposal.
2) PRIDE: have you ever heard someone boasting he’s driving a Peugeot? Nope! People who drive Ferrari do! Same here! It’s always more delightful to awe somebody by saying:” I have a 700 year-old sword” than: “I have a 70 year old sword”. The worst of the shame being : “I have a Chinese wallhanger!”
3) SLOTH: let’s face it, Koto swords are easier to study than Shinto swords. Learning about 5 traditions is always easier than learning a multitude of traditions.
4) FAKE NEWS: everyone knows Kamakura and Nambokucho swords were perfect! No, really! They had beauty, grace, efficiency... wait? Efficiency? They were so practical and efficient that they had to be suriage during the Sengoku era to be useful in battle.
5) SIZE MATTERS: Everyone knows Shinto swords are useless because they weren’t made for battle but to display showy qualities. Not so with Koto swords! Eeeerrr! What about those Nambokucho swords where samurai from the Southern and Northern court fought to see who had the longer one? I mean the sword, right? Isn’t that being showy?
6) SCARCITY: actually, this is my only serious argument, so no joke here. Most of the swords on the market today are Shinto swords. It’s logical. After all, they are more recent, didn’t experience real battle, so it is normal that we find an array of swords ranging from bad to excellent. Now, the early Koto swords are more scarce. It’s logical too. They’re older so had many more opportunities to be lost, destroyed, whatever. So it’s also logical to postulate that most of those that survived were cherished family swords of quality. If this is the case, we can also postulate that many swords of lower quality disappeared in the maelstrom of time, leaving only the upper quality swords. Thus, those Koto swords we have are truly better than the more recent ones because the good ones are the only one left!
Once again, anyone feeling piqued by my post, please read the disclaimer and realize this was written with a very humorous effect in mind! Please don’t send an evil ninja to exterminate me!
In the end collect what you like and attracts your eye until you stumble upon your own personal treasure.