I think we might have played a major part in the extinction of the irish elk, but most of the giant mammals of the period were simply deselected by nature due to massively changing temperatures, and in particular the big cats being the apex predators were lost through a major loss of prey species. The recent and excellent series "Lost land of the tiger" shows just how important a really diverse ecosystem and rich prey structure there needs to be for creatures as energy expensive as big cats.
Places that mirror tundra areas of the past are still remaining in north america and canada, asia and russia, and they still retain a selection of major predators, but they are still very much smaller than the major predators of the past, and often more generalist in their habits. Most bears for example remain, and have evolved but are fully omnivorous, and polar bears which tend to be predators of marine life are showing signs of environmental stress the cats that explore the colder places of the earth are again at the edge of their range. Pandas too are struggling not only because of the human factor, but because they too have become a little over specialised.
The britain of the ancient past was a place that was once attached to mainland europe, and in many cases its wont be a case purely animals dying out but simply relocating back to mainland europe gradually, and in turn expending their range increasingly east and south , and as they did so evolving to become new species, and of course new species outcompete older species and the wheel of evolution turns.
TBH though I too think of the loss of some species as a shame, after all who wouldnt want to see a similodon, a terror bird, or a mammoth, or any one of the ancient elephant, rhino, or giant sloths with their staggeringly different body shapes, but it is a trade off. For many of them were as transient as many modern creatures must also be, and which is more beautiful, a sabre-tooth or a modern tiger, the staggering size of the mammoth or the intense advanced emotional state and intelligence of a modern elephant? Nature must after all perfect itself and progess onward. Modern animals carry the characteristics of their forebears within them, in the genetic code.
In addition despite our human ancestors being tough, there were places that would have been difficult for them to survive in, and there were some big predators that were much more capable of surviving in intense cold and would have exceeded their range, so I'm personally pretty convinced that were were not soley responsible for major animal extinctions, it was just a fast period on climatic change that made things difficult for anything that didnt have generalist feeding and living strategies.
Overall though, I think we are very lucky. Not only can we enjoy the simply glorious selection of modern animals as the cutting edge of evolution, but also their ancestors, we can compare their differences , their flaws , and follow the map of changes that lead to either success or failure, and through our comparisons learn about modern life on earth, and appreciate it at a deep level of understanding, and in context the environments of the past.
This way we can add to the reserve of learning that helps us protect existing species adequately, giving creatures space enough, diversity enough, and ecological richness enough to prevent their genetic diversity becoming so small during times of radical environmental change that species are lost unecessarily. Our past can be used to protect our future. We have many of the lessons we need to prevent extinctions demonstrated for us in the fossil record.If the neanderthals and early humans could cause extinctions of ice-age species, then that power is as nothing compared to the power we have now as modern humans to cause much wider spread extinctions. If we mourn the loss of creatures past we should remember also that we are now exterminating a greater variety of creatures than we have ever done before at any point in history, and the only thing more threatening to life on earth than humans is basically a meteor strike.
But yeah, if we could bring back the creatures of the past, it would be a truly amazing thing to see.
