What you’re describing sounds like normal camping, you hike for a few hours and pitch your tent somewhere. Isn’t the drive in picnic variety just an American phenomenon?
No. "Camp sites" where you drive in and stay in an allocated spot in your tent or vehicle are definitely common in Europe. They have washing facilities and often much more: playgrounds, restaurants, even water parks. Some people also hike and camp in the wilderness, but I'd bet it's a minority.
I'd be fascinated to know where you live that "hike for a few hours and pitch your tent somewhere" is normalized and the very idea of a camp site sounds foreign.
In Australia there are a massive variety. At the easy/luxury end there are camping-parks where (as you describe) there are playgrounds, full washing facilities, often an on-site shop and cafe, usually chalets available for hire, pitches for your RV, and preallocated tent-pitches. At the more basic end there are sites in the national parks where you put a fee in an envelope on your way into the park, then drive your 4x4 or hike to a really remote spot and set up semi-wild.
There's usually some sort of toilet facility.
I'm not aware of fully 'wild' camping here, and you'd want to pick your spot carefully because of the crocs and snakes. Not that they're fully absent from the marked places!
Can confirm wild camping in Queensland. There are laws that prohibit camping on private property without permission, but there definitely non sanctioned camping going on in the norther states.
I mean sure, here in Sweden there are big camp sites that are essentially grass parking lots where people squeeze in their camper vans or sometimes even tents like sardines. They have amenities, maybe even a mini golf course!, but that's a completely different thing. Nobody would compare that to actual camping, and most people think it's crazy and make fun of it.
But what shocked me in the US was that there are "nature" camp sites where you more or less drive up to a designated, single tent spot where you pitch your tent. It's baffling and seems to entirely defeat the purpose of camping in nature.
In the UK at least (apart from Scotland), wild camping is only legal with permission from the landowner (even in national parks), so most people use campsites to avoid the possibility of being driven off by a ranger or angry farmer.
Same in Germany; actual wild camping with a tent is not allowed. However, there is some provision to allow sleeping in the wild (sometimes called "biwakieren", using a bivouac or similar basic shelter; e.g. a tarp).
When my SO and I did a multi-week cross-country trip we went for whatever designated camping sites was close by and pitched our tent there (except Berlin, there we went for a Airbnb). Though for us this mostly replaced switching hotels every other day. We then did local stuff around there, like canoeing, hikes, museums,...
However, we did spent some nights in the wild, with camp beds (US: cots) placed next to the car. But when you don't know the general area, finding a good spot for that proved rather difficult. Entering forests with a car is usually verboten, and close to main roads there will at least be a lot of morning traffic; OTOH we had a pleasant conversation with some elder locals on a morning hike when we found a spot near a nice fishing pond in a small forest (yes, yes, driving there was erlaubt, not verboten). They only regretted not having some coffee for us :)
If you're considering going to Scotland and are thinking of that sort of thing, you should also look up the concept of a "Bothy" or "Bothies", which tend to be in mountains. They are a sort of refuge/simple transient accommodation. Generally small, quite basic huts in the deep countryside and remote places, where you can stay without booking and for free, but just overnight while you're on your way around.
The key is in the word "nature", lots of people enjoy being in more or less wild nature. And camping, wild camping in American, is about self-sufficiency and simplicity. You only use what you carry on your back, and you don't bring more than you absolutely need.
I'm not saying you shouldn't put your tent on a crowded lawn, or next to your car, if that's you want to, but it obviously doesn't serve the same purpose as hiking up the mountains and putting your tent next to a brook, miles away from anyone else.
Bivouacs/shelters are indeed quite popular, there are lots of permanent ones made of wood out here. Basically just a slanted roof. https://vindskyddskartan.se
Obviously rain is the big problem with not even having a tarp, and as tents are generally heavier than a tarp I'm not sure why that would count as the easy route.
Tents feel more like a house, you're sort of indoors. Plus it holds itself up, so easier to set up than a tarp, unless you just use the tarp as a blanket. I tried that, in the rain, not recommended at all.
