You can do much better than that software. VTK has lots of tools for surface mesh manipulation, path search between control points, etc., so for example you don’t need to specify a dense set of control points (that are very tedious to edit).
In 3D Slicer we added very powerful widgets for curve editing, with the option of constraining the curve to surface, insert control points, resample control points, etc. We also added a dynamic modeler module that allows you to cut surface patches out with closed curves. See a short demo here of what you can do without any programming (and you can customize everything, simplify the workflow, the GUI, etc. using Python scripting):
If you must do everything in a web browser then you can set up a cloud computer, or port all the missing features to vtk.js. Probably some of the VTK algorithms are already there, but you might need to invest time into getting sophisticated widgets.