Even in the UK there's plenty of wide open countryside.
You mean there is a few miles between your towns. I never made it past the Heathrow airport, so I cannot comment on English country-side having never been there. However, as a general rule if you can walk between towns in under a day, then you don't have country-side. I have been all over Germany, Austria, France, Spain, and Italy and other than a few small designated forested areas, they have no country-side. If it is not occupied by farms, then the towns and cities spill into a sub-urban sort of life-style.
The only place I can achieve that in the US would be someplace like Los Angeles, where I can walk from Hollywood to Beverly Hills to Santa Monica in less than a day, because they also do not have country-side in the greater Los Angeles area.
In case you were wondering, this is what I mean by "country-side" (this is my backyard):
If Europe was as densely packed as you claim how would we be able to grow any crops?
You wouldn't, at least not enough crops to feed your population. Which is why Europe has to import a lot of its food. Much like Alaska, although we import our food for other reasons, and not because we are crammed together like sardines. Approximately 93 million tons of food has to be imported from outside Europe by EU member nations every year, and that number is increasing as the population of Europe increases.
East Anglia (the bulgy bit on the east of England) where I live is called the breadbasket of England and has loads of farms and wide open fields.
Scotland and Wales both have vast areas of stunning wide open countryside to explore and enjoy.
Yes the towns and cities are densely packed but we have extremely strict planning guidlines about not building on green field sites and we have some large national parks that get visitors from all over the world.
You and I have very different definitions of "vast" and "large" and farmland does not count as "country-side." However, I will grant you that England is better able to feed its population than other European nations. It probably goes back to WW II and the shortages that occurred. As I understand it, there was a serious effort to get English citizens to grow more food and harvest what they could from the land.
What you consider to be a "large" national park is tiny by comparison. Just one of our national parks in just one State is the size of Wales. Wales = 13,298,560 acres, Wrangell-St. Elias National Park = 13,175,680 acres.