I appreciate your willingness to express your wish thoroughly and eloquently.
But as you have correctly predicted, I am going to tell you that it is simply not feasible, at least for CVX. I would encourage you to check out YALMIP. Johan Löfberg, the developer of YALMIP, has said that I’m the “disciplined” guy, and he’s the “undisciplined” guy. He has indeed taken efforts to support a wider variety of models. A lot of people swear by YALMIP; indeed, CVX was inspired in part by YALMIP, so I’m a fan too. But the complexities required to support the wider variety of models are simply not something that I can afford the time to take on.
Looking at the bigger picture, it is certainly possible for someone to build a system that “handles”, in some sense, a wider variety of models than CVX or even YALMIP. But I cannot think of a single piece of the essential CVX code that could be re-used in such an effort. CVX was built for the ground up for DCPs, and it informs the design throughout. So even if I personally made the decision to move forward with an effort to extend support for nonconvex problems, what I would be moving forward with is an entirely new system.