Why would it need to be a button? For that matter, why would it have a C on it?
That would make sense, so games that rely on the mouse functionality would still be playable in handheld mode. You could just use the touchscreen, and I assume that's what most games would do, but specifically for camera movement, that wouldn't really work (I mean, the DS and 3DS tried, but it was kind of awful, you wouldn't be able to compete in any sort of competitive shooter if the other players weren't also using the touchscreen, it would be worse than using the right stick)
Taking a page out of Valve's book (with the Steam Controller and Deck), seems like a good idea. As long as they don't get sued. Or maybe it would be good if they got a taste of their own medicine.
It's waaaaay too tiny though. It's probably usable, but doing quick long movements would be impossible without really cranking up the sensitivity, which then makes accurate movements impossible without using the gyro, which is the same problem we have with the right stick already, so it doesn't seem like it actually solves a problem... Might be slightly better than using the right stick though. I'd probably use it in single player games. But really, it needs to be at least bigger than my thumb to be much good. If you use it like you would use an analog stick you're going to move your thumb off the tiny surface and then it won't register anything.
There's one thing that doesn't make sense about that explanation though. Since even a child's thumb is bigger than the button, it would effectively be a single point of sensing. So why does it need to be a square? It seems like a circular touchpoint would be more intuitive to use. It just seems really arbitrary. I guess there doesn't need to be a reason for it to be a square, the capture button is also a square after all. Maybe they just didn't want it to look (and feel) too similar to the home button so people would confuse them. But now they run the risk of people confusing it with the capture button instead. That could be why the C was removed, actually.