Having lived in the UK, I'm not sure I'd agree with that assessment. While in the US we are limited in how much game we can take by game regulations/seasons, there are a lot more hunters, and a lot more game to hunt here in the US than the UK IME. In fact, I was stunned by how...sterile...the fauna was in the UK. Nowhere near the diversity (or density) of game as in the US. Which makes sense when you consider the sheer difference in size of our two countries, plus the fact that we haven't been killing stuff off in our country for anywhere near as long as what has been done in the UK.
Case in point: While I only bag one deer a year (mainly due to time and location), there have been years where I have shot well over 600 prairie dogs in just one week, never mind that when deer hunting, it was a decision of which deer of the 60+ I was looking at, that I shot. Never mind dove, quail, pheasant, wild pig (5 varieties in my state alone), duck, geese, coyotes, bobcats, fox, turkey, elk, bear, bighorn sheep, pronghorn antelope, javelina, tree squirrels, rabbits etc...
Then add all the imported exotics (which in many states are not limited by law on bag limits and seasons) like axis deer, auhdad, and others, and you have a much denser wildlife population in the US. Hell, there's more whitetail deer in this country now, than when Columbus landed back in 1492. And then there's the declining hunter population here as well, but again, we're a country of 330M people. Sometimes I think people forget that the UK is under 94 thousand square miles big (with ~67M people), and the US is 3.7 million square miles. Which do you think is going to have a more dense wildlife population? Especially when so much of our wilderness is part of the government park system, and many parts of it are largely untouched and in pretty hospitable/temperate places (Yellowstone, Glacier, Gran Tetons, and those are just some of the big, well known ones).
I mean, come on, the state of Montana is almost 50% larger than the entire UK.