It was released before the New 3DS, so the only way for games released before the New 3DS to have controls using two control sticks was to implement the Circle Pad Pro. This matters, because trying to adapt a design. meant for two control sticks onto a system that only has one circle pad requires making changes. (If I recall correctly, the Strobulb from Luigi's Mansion 2 was created to get around only having one control stick.
) Admittedly, most of the time the second stick is used for camera control reasons, so in theory the accessory is largely optional even in the game's its compatible with.
There's also Kid Icarus Uprising, a shooter that would normally use two sticks on any other system, but instead uses the touch screen as the second control stick. This works for people who are right handed, as they can use their dominant hand for the touch screen and their non-dominant hand for the Circle Pad, but it doesn't work as well for left handed players. Left handed players would use their dominant hand for the touch screen and their non dominant hand for the face buttons. But the face buttons, A B X and Y, aren't really equivalent the the Circle Pad because the buttons can't input every direction and can't vary in magnitude. The Circle Pad Pro does fix this problem by allowing left handed players to use a Circle Pad on the system's left side.
Apparently, the Circle Pad Pro was announced for Monster Hunter 3 Ultimate, so I assume its integrated really well into that game. But I know nothing about Monster Hunter so it's hard for me to say for certain. Seems like it was also for camera use, but the non-Circle Pad Pro way to control the camera is also the touch screen in a game with a customizable touch screen. So using the Circle Pad Pro would save space on the touch screen to use for other functions.
I looked at a reddit thread to see some testimonials and one person said the Circle Pad Pro made holding a 3DS more comfortable by increasing its size. Considering the number of people who have problems with the size of Joy-Cons, it's possible this is a very real benefit that isn't noted on the internet that often.
The thing killing me here is the price though. In the U.S the Circle Pad Pro retailed for $40, which is more expensive than a controller really should be. Worse, a number of games that have functionality for the C Stick on the New 3DS didn't also add support for the Circle Pad Pro, which surely annoyed anyone who did like using the Circle Pad Pro and makes the price more annoying.