If I remember correctly, Homo sapiens sapiens was not only coetaneous with Mammoths, but we are widely considered to be one, if not the main cause of their extintion.
Also constructions like Gobekli tepe, with it's carvings and decorations, predate the extintion of Mammoths by something like 6000 years.
birds are the continuation of the theropod dinosaur lineage.
humans are the continuation of the early synapsid lineage also present at the time (which later gave rise to the early mammal progenitor).
when people say birds are dinosaurs they mean the lineage didn't branch as much as it did for humans, which I think is more survivorship bias than anything.
People say birds are dinosaurs because every living thing is in every clade of it's ancestors which means they are dinosaurs. They're also a lot of other things from all of the other clades so they're not saying that birds are just dinosaurs, but that they are part of Dinosauria and every other clade of their ancestors and so too will all of their descendants be.