Freiburg in Germany. The black forest literally starts at the edge of the city. It was a while since I was there, but we definitely walked from the city centre to the forest. Might have been 30 minutes. Would be quick on a bike.
Zaragoza in Spain. Walked from the old city centre to the edge of the city (which was apartment blocks) in a reasonable time. It was mostly agricultural land, but I recall
some forest along the river.
San Sebastián, also in Spain. You can walk into the hills, but we went to the beach.
Parks covered in homeless people is a US problem, in my experience. I've been around most of Europe and haven't seen anything like that. The homeless problem in the US is a policy choice, just like the urban development is. It's easily solvable given political will.
Good for you that you have the money to buy acres of woods (or an exclusive parcel bordering woodland) and a job that can support living in a remote place. For many people that would want that lifestyle it's simply unattainable, and we should give them reasonable alternatives.
I live in Prague, my parents in a nearby town. Walking 10 minutes gets me either to a metro station or an "urban forest"/heavily wooded park called Stromovka. I think it's a very appropriate place for kids to play in nature. My parents have walking/biking access to similar places.
Not every place in Prague is this close to nature, but it's common enough that someone seeking that kind of housing can get it at basically no premium.
I have acres of woods _in my backyard_. There are also no homeless people sleeping on the benches.