That's perhaps true of some species of deer, but moose, elk, caribou are all old world species that are relatively new. Brown/grizzly bears, wolves, foxes are all recent arrivals across the land bridge.
I got news for you. All the common animals today don't just consist of moose, elk, caribou, brown/grizzly bears, wolves, and foxes.
And you still seem a bit confused on your entry dates of species into the New World. Wolves entered the New World some 1.8 million years ago. The Gray Wolf did come across, but the Eastern Gray Wolf developed here in North America along with the Coyote between 150-300 kbp. In short, they were already here when humans arrived.
http://www.searchingwolf.com/wevolve.htm
As for the moose being new and unique to post-Pleistocent New World humans, that isn't really the case. Modern moose (
Alces alces) basically replaced the stag moose (
Cervalces scotti) in terms of being the big deer on the continent.
And those pesky brown bears were already in place and waiting to eat humans when humans started really moving into the New World. This is because the bears moved in at about 26,000 years ago, ahead of most humans.
Yes, pre-Pleistocene, Pleistocene, and post-Pleistocene New World taxa and environs were all different, no doubt about it, but most of the common animals around us today did not come across the land bridge during the Pleistocene. A few did, but not most as you stated.