I prefer to check the barcodes myself, and so far it looks like the Neanderthals have the right kind of barcodes to be considered Europeans!
Both the mitochondrial and nucleic DNA evidence agree that they are clearly not human, not nestled into that chart somewhere.
The chart is based on genetic analysis, it is not wrong in representing what it is, a measure of genetic distance.
It agrees with other studies on DNA genetic distance between human populations and that distance will hold true independent of what neutral DNA markers are used.