We have animals now that are closely enough related to extinct animals that we could conceivably bring them back through incremental genetic modifications.
For example, if we wanted to bring back Direwolves, we could edit the DNA of a gray wolf maximally such that it can still be fertile and be born by a non-edited gray wolf. This would give you a wolf closer to a Direwolf and then you could repeat this process using each previous generation as the carrying mother for the genetically modified embryo.
If you were to clone a New Yorker thousands of years from now and put them in an exhibit in a Zoo marked "New Yorker", lots of people would be very impressed, even if they didn't start cooking New York-style pizza.
I don't think that's true. We raise animals without their parents all the time, both on farms and in zoos. The animals we end up with are still recognizably horses or rats or chimpanzees, with unique behavioural traits that could be studied for years. Even if you lose the social structure of mammoth society, you would still be bringing back something fascinating.
Animals that can be hatched from eggs have even less dependency on their mothers, too.
I have 16 chicks in the brooder that have never been around a chicken. They never will be around any but their 15 "siblings."
They're still acting like chickens, and they will continue throughout their lifetimes.
And that's not even getting into the complex interaction between a mother and her unborn child, including cytoplasm, mitochondrial RNA, etc. It's not like an organism springs whole cloth from DNA alone.