You do not believe that it is a tiny bit illogical to claim that a debate about whether modern humans and Neanderthals coexisted together in the Mediterranean and Middle Eastern region 60,000 years ago or not is actually throwing doubt on whether modern humans existed 6000 years ago???!!!!
Radiometric dating have a +- 10% or so error at very early times. So 60,000 years can be 55,000 or 65,000. Hence the extent of overlap between the two populations during a 10,000 year window of coexistence is being disputed as we need a bit more dating work to reduce the uncertainties.
However we know that modern humans existed 100,000 years ago and Neanderthals also existed by 70,000 years ago. The claim about 6000 years ago they existed or not is just absurd.