Cloning is so far away, that with any luck, General AI beating human intelligence will become available sooner - So both Homo Sapiens and cloned humanoids would be "lesser intelligence" subject to whatever "greater intelligence " (GAI) will decide is good enough for both.
You are trusting journalists too much, or watch too much TV. Your assumption is wrong.
Cloning humanoids is way off in the future, we are not near close, even with simpler and much better understood mammals.
Even cloning simpler animals like sheep Dolly is extremely complicated. 95% fetuses are not viable, and most of the few which are have severe diseases causes by wrong genetic signals. Dolly has arthritis from age 4, and has to be put down.
And in this situation, we had really good understanding of biochemistry, perfect fresh DNA, fresh egg to put in into, and real sheep mother to carry the embryo to term - we would have any of those for Neanderthal, so there are many more things to go wrong. So rate of success would be much lower, you will need to engineer many thousands of embryos to get one viable.
Cell division during pregnancy is complicated biochemistry dance, where cells send chemical signals to each other to guide proper formation of organs in neighboring cells. Slightest mistake in signaling, and organ is misformed, and embryo is not viable, or has some disorder.
And those are only technical obstacles - there are many more ethical obstacles, as you correctly noted.