Common genes do not imply cross-species (human/hominid) breeding

We share around half of our genes with the ordinary banana...
That assuredly does not come from humans BREEDING with bananas...  It most liikely comes from humans EATING bananas, pretty much forever, and probably throughout the universe and not just on  this planet.
Likewise, the first experience humans ever had with Neanderthals on Earth was watching friends and family members being killed and eaten by them, so that eating a Neanderthal that had been killed in some battle would have just been sending the Neanderthals a message in their own language...
In both cases, what you seem to be talking about is bacterial insertian of genes.
There is a claim that, because some humans have a certain small number of genes in common with Neanderthals, that humans and Neanderthals must have interbred. That amounts to thinking that a Neanderthal male could/would rape a woman and, rather than cooking and eating her afterwards as usual, somehow or other keep her alive long enough to bear a cross-species child, raise that child to reproductive age, and have him/her breed back into human populations without anybody catching on, i.e. the claim is ridiculous.
In real life:
Neanderthal females would kill that woman the first time her new owner left her alone for ten minutes.
The woman wouldn't fare any better than the subjects of the commie attempts to breed humans and apes into super workers in the 1930s.
Humans would notice the child was different (really different...)
And humans would kill that child and everybody else like him as part of the same program which killed out the Neanderthal. They would not need DNA tests to determine who to kill for that sort of reason, it would be exceedingly obvious.
https://youtu.be/mZbmywzGAVs
In other words, it would be a miracle for something like that to ever have happened once while the claims from Paabo et. al. require it to have been going on all the time. That is, for human/hominid cross-breeding to have left detectable traces in the DNA of modern humans, it would have to have been entirely common.
One zero-probability event in the history of the universe? Maybe, but not an infinite series of them, i.e. not something that stands everything we know about probability on its head.