This is actually really important and comes up in Anthropology classes all. The. Time.
As long as homo sapiens have existed, we have had the same emotional and mental capacity as you and I do today. You nailed it. They were US. Even Neaderthals existed alongside and had offspring with Homo Sapiens for many thousands of years.
There's much evidence that cavemen would have had complex spoken language, culture (learned information passed down), symbolic interpretation, and I think they most certainly would have been able to handle holding a baby. In fact I have my suspicisions that an ancient homo sapiens mother may be a more present, attentive, and knowledgable mom than I could be today.
Do not let media trick you into believing we are the pinnacle of humanity. Unilinial evolution theory (google it quick I beg) is BUNK, GARBAGE, and the root of so much evil.
We've been human for a long, long time, and we are not inherently better than all those who came before.