I would just like to state that I am one person who enjoys trying to get to cubeshape for curvy copter. I think it is a nice puzzle to try to be able to get to cubeshape "cleanly" (without any even partially forcing moves), and while it feels like a goal that is entirely achievable, I'm not really there yet. Sometimes I do solves where I just don't worry about how long it takes so that I can try to go completely without forcing moves, and it takes many minutes to get to cubeshape. I think it is a very interesting part of the puzzle. (It is not surprising that I enjoy this part of the solve; I also really enjoy solving truly scrambled Magic and Master Magic puzzles, and would love a competition event which involved unscrambling scrambled Magics and Master Magics, if only a scrambler could be built for that.)
However, that being said, I recognize clearly that it is not very practical for competitions, and especially not for official competitions, since forcing moves are so easy to do on curvy copter. I can see where the practical thing to do, certainly for any official competition, would be to do jumbling moves but keep it in cubeshape. As for here, I wouldn't make any changes without the usual end-of-year polling process, but I could see considering that change if community opinion were overwhelming in that direction.
As for displaying the jumbling moves, I've had the intention for quite some time of eventually having optional display modes for various scrambles, and one of these would be for curvy copter. The idea would be that the scrambles would be identical positions, but given in different notation. For curvy copter, I would probably add the parentheses as one option, and use some of the shortcut move notation as another option. Also, for 15 puzzle, for instance, there would be an option to just give the final position, for people with magnetic versions of the puzzle. This is something I still hope to do as one of the next improvements, but right now I'm rather busy so it might be a few more months before I can get to it,.