Mammoths dies out as the result of all the ice melting, the got too hot and died. So as for them finding a niche in the tundras here's why that won't work
1. The tundras are getting hotter, they'll end up just dying out again.
2. They'll have to compete with current wildlife for food and there isn't an animal in the tundras today that could compete with a heard of mammoths.
3. They won't be prey for the same reason. So it will work like this: Mammoths eat vegetation, other herbivores have no food and die out leaving only mammoths and a few other species. Carnivores are then left with primarily mammoths and each other as prey options. The carnivores will then kill each other off because they can't take on the mammoths and eventually only mammoths will be left and a whole ecosystem is dead.
As for it not being unnatural. It is very unnatural. Mammoths (still using your example) had their shot and couldn't cut it, so they died out. It's called natural selection and without it we wouldn't have ever evolved and nor would any other species of animal alive today. There is nothing more unnatural then reintroducing a species that has already died out because of natural reasons. Reintroducing an extinct species is no better than the Cane Toad crisis here in Australia, and would probably be even more disastrous since at least Cane Toads have predators. I wouldn't put the reintroduction of extinct species on the same level as some of the other things humans have done to the environment but it is in the same league.