Does this 3D rendering method exist? How is it called? Is there any C++ algorithm for it?
Yes, such rendering exists. It is called a parametric surface and there are several methods you can render such entities. In this case you are talking of bezier patches.

Turn the object into enough triangles. The triangilation is called tesselation or meshing depending on what literature you read.
There are lots of applications that do this allready. Most cad applications work this way and many graphics apps like blender can work
this way.

raytrace or raymarch the surface in a shader.

Microdice the surface

etc.


The reason this is not often done is that the on line tesselation is for most intent and purposes just a hindrance to asset developpers. You get the same with subdivision schemes and its just easier. The primary reason to drag the bezier (or nurbs, or t-splines...) is that unless you need to find better intersection of underlying primitives then it is often just a waste of resources. Although subdividing your mesh  via a subdivision surface is allready more useful