"The horizontal component of lift is equal and opposed to the horizontal force at the stabiliser"
Where are you getting that "horizontal force at the stabiliser" if it is not being generated by the very sideslip you think you have made go away. Fred you can't make it go away else you will have a strong tendency to become a frisby.
Convinced yet?
Milt,
as you deflect the rudder, you turn the fin into a cambered airfoil. Thus it generates a force without any sideslip.
Coincidentally, that behaviour is also required if you are ever to get out of zero-slip flight using the rudder. I don't think we need to debate whether that is possible?
I shall continue waiting.
Regards,
Fred