I'm not saying predators would normally decimate prey populations left and right (in fact the success rate among wolves is all of 8%,
when they decide to pursue ungulates in general as opposed to other prey, i.e. rabbits), I'm saying that when they do manage to avoid antipredator strategies and take something large, it isn't pretty. It's worse for prey species like rodents and lagomorphs, which not only support the bulk of the food web but host a large diversity of diseases and parasites as well (just google the phrase "squirrel bot fly").
Now for the scientific tangent/wall of text...
The actual health of the ungulates depend on the life history of the ungulate species in question, the habitat, predator density/extirpation, and ungulate density. I'll use stats for Pennsylvania as an example, but the same phenomenon occurred throughout the eastern and midwestern U.S.:
The pre-colonial habitat equipped with both a Native American population and several healthy natural predator populations (eastern cougar, eastern timberwolf, red wolf, and black bear) could support approximately 14-16 whitetail deer per square mile sustainably without a negative impact on the deer population size. Because the deer population was kept in check with what the habitat could provide, does had no problem finding readily available nutrition, which resulted in two things: 1. they often became reproductively mature at 6 months old as opposed to 18 months, able to birth a single fawn during their first pregnancy, and 2. they gave birth to twins or occasionally tripletts every year after. Rates of disease were low because the density was low enough so that individuals rarely came into contact with eachother outside of the rut and the occasional bachelor herd.
Beginning in the mid-19th century most of the state had been deforested (most of the native trees were white pine, which were used for naval ship masts) and by the early 20th century all of the native predators had been extirpated with the rare evidence of a black bear being the exception, and deer and wild turkeys were nearly extirpated (if you saw a deer track walking to school you ran back home and grabbed your dad and neighbor to come look at it). Market hunting didn't help the situation either.
In the 1930s market hunting was banned and a lot of farm land fell out of use, which allowed the forests to begin to regrow. This reduced deer mortality, and all of a sudden deer had access to a predator-free utopia full of food within reach (seedlings, saplings, and shrubs growing in the understory) for the next 40-50 years. The patchwork of forest regrowth and farmland was able to support an average of 70-80 deer per square mile, which caused massive crop damage, an untold of number of vehicle collissions, and the eventual rise of diseases such as chronic wasting disease, epizootic hemorrhaging disease, increased mass kills from botulism and anthrax, etc.
As the forests matured into the 1980s, suddenly there were more deer than the habitat could support (120+ per square mile in places like Gettysburg National Park, which didn't and still doesn't allow hunting because the land is federally owned), which led to massive habitat degradation. They literally ate themselves out of house and home and changed the composition of the forest, which is now largely devoid of palatable species such as the sugar maple and dominated by unpalatable species like black locust and black cherry. The understory has been reduced from a diverse plant community to a carpet of unpalatable hay-scented fern.
The lack of nutritional availability let to delayed reproductive maturation, smaller fawns, an average of a single fawn produced per year per doe, and an increased fetal resorption rate (deer undergo delayed embryonic implantation, meaning that a fertilized egg only implants and develops if the doe has enough fat reserves, and then the embryo can be resorbed if conditions aren't favorable in the spring). The habitat has been degraded to the point where it can only support 10-12 deer per square mile on average, and it continues to degrade in spite of the fact that disease, starvation, road mortality, and increased hunting incentives (the number of hunters in PA has declined since the 70s) and eradication programs have reduced average densities to the 40 per square mile range.
Every February the PA Game Commission in cooperation with the state university system (Penn State included) conducts stream-walking surveys to count the number of parched dead deer with nothing in their guts but brown pine needles and dead grass to get an estimate of regional overpopulation.
I would argue that this is an extreme waste of several resources because the degraded habitat impacts all the species in the habitat, not just the deer. The habitat still cannot support a predator population and will not in the foreseeable future, given urbanization. Right now their predators have 4 wheels and a windshield.
Reducing densities to an average of 10 deer per square mile would allow the habitat to recover to the point where it can again support densities in the 15-20 range, and given that we can expect twice the birth rate due to increased nutritional availability, hunters would be able to harvest the same number of deer as they do today with a density of 40.