Also, 'continuity of consciousness' is just a specific sensation that this specific clump of atoms has at each point in time

Or not. Memories are genuinely lost, if someone makes a Horcrux and then dies some years later. Moreover, according to the Defense Professor in snake form, the maker's personality could also change due to influence from the (two) victim(s). The result need not act like the maker at time of casting would act if placed in a new environment.

See also major's point.