I am not entirely sure that it is technically possible to design a basepoint snapping that is compatible with gizmos-driven axis restriction in the other way.
For example, C4D tried to design something like that, but this attempt resulted in well known cumbersome and overcomplicated system with much lower usability and discoverability which also require a pivot point system similar to Max and Maya (which doesnot exist in Blender or AutoCAD and bringing which assumes quite deep rewriting nearly from scratch since influence object’s structure).
As a result closest and basepoint axis-restricted snappings in C4D are rather challenging to perform.
