CVX can not solve all convex optimization problems. Please read Why isn't CVX accepting my model? READ THIS FIRST! which was linked in an answer to your previous question. But with the benefit of reformulations, as seen in many posts on this forum, CVX, aided by solvers it calls, can solve many convex optimization problems.
For many of the problems which can be solved using CVX, there are more specialized algorithms which are faster and/or require less memory. Also, the transformations applied by CVX, “modeling time”, take time. CVX is designed to save human time and effort for the modeller, but doesn’t result in the fastest executing code.