Berlin has all of those,(albeit it is one of the highest open space cites there are) hell London has over 20 skateparks that is just for skating that is before you consider a lot of the Thames bank serves as open parkside. With the only traffic pedestrians and boats on the water. High density doesn't have to mean no space, it does have to mean no space for autos though, they simply take up too much space. If you have mass transit that reduces carparks and roads to say 10% space opens up a lot more. Playing in the streets or cafes using sidewalk for tables is a lot more compatible with trams running on fixed rails and electric overhead than with massive auto traffic.

High density doesn't me

Why are you saying "no space for cars"? You need to build multistory and underground garages here and there and you are all set.

Because that is a fundamental point that seems to often be missed in density discussions. Cars are great for low density areas and appealing at high density areas where as mass transit is great at high density but awful at low.density this why park and ride stations outside the city are a thing.

Even if you could build miltistory underground car parks you still need roadway space. German narrow gauge trams quite happily chug Elon in two directions on 10 feet carry highway capacity traffic one highway lane is 12 feet. They don't need the space. So car traffic takes up much more space that means you can't have the same density. If you don't have those cars you could put all the factories and warehouse space underground and you would have even more surface room this would allow a higher density of people. To imagine this take a look at your closest city on Google maps and how much space is taken up by car parks and roads now if you can reduce that use to 20% while keeping the same city boundaries, the density of the city hasn't changed but a lot more land is available. How much bigger can your house be if the highway is only one lane wide? How many more people can you add in?

You would also have the issue of getting people to and from the miltistory underground but that is exactly what happens on car free areas. You take the tram/bus/train pick up your car/rental/rideshare/uber at the multi story car park and continue on from there. If it's right on the major highways around the area you reduce a lot of the traffic headaches.

Terrible not appealing, a synonym was autocorrected