They all branch from a common ancestor that originated in Africa and migrated out from there. Homo species migrated out of Africa LONG before modern humans, including the Neanderthal.
Our branch is relatively new and survived, the others died out. When inter-breeding was possible it apparently happened because traces of the other hominid species DNA can be found, but the species it came from is no longer around.
Do you understand that Neanderthal, Denisovan, and Sapiens are three distinct Hominim species?
Just so I'm clear, you state that Homo Sapiens Sapiens WITH Neanderthal and Denisovan DNA should be classified as the same species as Homo Sapiens Sapiens with NO Neanderthal or Denisovan DNA.
Do I understand you correctly?