Other things would rather quickly take their place. Some bears split off to become either mainly or entirely herbivorous in some areas, taking say, Elk's place there. Or maybe a mooses place, assuming they are included in "deer"
Cougars' and wolves' numbers would decline dramatically, maybe to extinction n some places. Where they do survive they would become smaller, and travel in smaller groups. That would probably make the undometicatable. If humans domesticate before this happens, then many North Americans would probably selectively breed dogs more, leading to a greater variety of dog breeds. Some would be more geared towards working, eventually leading to the pet dogs, while others would be bred to be like sheep or cows or probably most like pigs. They would keep a group of them, and feed them the scraps of other kinds of animals fattening them for the harvest.
Pronghorns and stuff may spread North, forming large plains communities.
Maybe south american creatures fill niches. Peccaries fill the roles of deer. this could cause jaguars to follow them, taking a place as a prime hunter in the southern US and in Mexico. That is if the wolves have already primarily retreated to the North West and East.
Caribou could reach in from the north, probably reaching Upper New York State or so. That is assuming that they aren't included with the deer family.
Native Americans would probably be relatively confined to rivers and seaboards, because the Pronghorns of the plains wouldn't provide near the amount of food as a bison. Eskimos or Eskimo like cultures would probably be dominant in the north, reaching maybe to Long Island.
Natives on Rivers would have a benefit relying on fishing and breeding sheep/cow/pig dogs, they may be some of the first cultures in the world to develop farming, assuming they could get their hands on either south/central american crop packages or that the OTL deer did indeed wipe out some plant(s). They could develop relatively on par with everyone else, the real split in tech coming once Asia and Europe start to actively trade, the Americas would probably fall behind.
Once Europeans reach the Americas they find relatively successful empires clinging to waterways. The empires are advanced enough and populated enough that Europeans can't do a lot of colonizing. They instead to turn primarily to trade, in the process exchanging diseases. The Americans would have the problems they had OTL, but so would the Europeans, who were still in recovery from Black Death.
Well, that's it for my wall of text.