A wave of certain Eastern mysticism and the idea that due to the relatively complicated forging process the katana was of amazing quality.

Basically.

It's not a long, supported, sourced answer but it's kind of the only one there is from a historical point of view. To explain the recent Western fascination with Japanese swords you might be better off asking a sociologist or someone who knows a lot about pop culture or something.

There is no real surefire way to measure the 'goodness' of a katana compared to a somewhat standard European longsword (or rapier, or sabre, or etc etc), because you get the classical problem of comparing two artifacts that are very much defined by their own individual context, separated by thousands of miles and sometimes hundreds of years, existing within very distinct environments. The katana was a good sword, for what it was supposed to be and do. The longsword or any other variation of European sword, was a good sword for what it was supposed to be and do.

Comparing really does not make that much sense unless one really delves into the specific fighting styles associated with them. And even then it's a weird excercise into what-if's.

One actual factual thing that might contribute to the katana legacy is that katanas were made with an edge of very hard steel, which allowed it to keep its edge for a long period of time and possibly made it a bit sharper than European swords. To offset that, the back of the weapon was comprised of a softer steel that gave the blade some elastic strength, so the entire blade is not too brittle. (<<== my understanding is that this differing hardness mostly comes from differences in the quenching process)

In contrast, European swords, because of a greater availability of good quality steel, were generally made of a uniform quality of steel. This allowed the blades to be a quite lot more flexible (European swords can bend, katanas can't. Well, they can, but you'll have to bend them back yourself...) and durable, but the edge itself was softer and more prone to wear and tear, comparing to a good quality katana. This difference in sharpness is first off debatable and second, very small if existant at all. The difference in hardness is factual, but can be considered somewhat irrelevant considering the notion that, if you were expecting your blade to get damaged during proper use, well, then that's what happened. Practically, it might mean, and I'm really wet-thumbing it here, that a skilled samurai could slice through a dozen more guys and have his sword receive a few additional nicks before it being re-furbished, but I mean, what practical difference does that make really?

So the factual difference is not much, but combined with the aforementioned mysticism and legendry about katanas in Western pop culture, things like that create these misconceptions. Especially when compounded by cool kung-fu movies and D&D rules.

edited for better syntax and spelling and expansion

Adding on to the question, is there anything in Lindybeige's Katana video that's way off base?

Katanas are brittle. Japanese steel is just shit. But in all seriousness, the iron to carbon ratio, was too high. Making swords hard, but brittle. Why? You always need flex, to distribute force being applied to the sword. The curved shape of the katana came with benefits, and drawbacks. The benefit was more protection to the wearer. The two drawbacks are, because of the curved shape, came mistakes during the making process. The other is that double edged swords can strike with both front and the back, losing this, limited the katana wearer.