They already do that. New subdivisions are required to keep up to 15% of land as open space (less, if they build recreation amenities aka "park") or pay a $400 fee-in-lieu (per lot) which can be spent on recreation needs.
The general idea is that once the new subdivisions start generating tax revenue, that some of this revenue will be reinvested back into the community in the form of local parks and recreation. So far, our taxes have gone up, but as a community, we have got nothing in return.
Your argument doesn’t address the fact that many older subdivisions were not required to have private amenities when they were built. At any rate, these private recreational amenities are inaccessible to other residents, in most cases, people who have lived here in Cleveland longer. I won’t be beating a dead horse today since there are many threads on here discussing the pros and cons of community parks and amenities, but a private little playground built only for those who reside in a subdivision, does not serve the same purposes as a larger community park. You may as well compare apples to oranges.
For reference our neighborhood is 25 years old, has roughly 100 houses on about 150 acres. But they didn't build any amenities and didn't set aside any land as open space. We had an HOA way back when, but it has since been abolished. DOT took over the roads many years ago. I enjoy parks quite a bit, we use the Neuse greenway a lot to bike. It would be really nice to have a greenway or a park somewhere in the 40/42 area that is closer. I have always thought it would be nice to have a greenway along Swift Creek. Starting up by Summerwind and continue down Barber Mill (maybe one day to Smithfield). Then maybe have a park or two that branches off that greenway, Maybe around the crossing at 42.
Most must pay the Fee because I haven't seen much of any Parks, or open areas, in most Subdivisions. A few have an area for Subdivision Owners only.
Actually most I see just set aside land as open space. Summerwind has 25 acres set aside for this. River Oaks has quite a bit, but since they are still building its hard to be sure how much. Kyndal has 10 acres set aside. Adams Point has almost 50 acres set aside. Broadmoore has about 20. It's just that usually this land is unusable anyway. So they just set it aside as open space to meet the county's requirements.
Just to clarify, AFAIK deeded "open space" must be open to the public -- not restricted to subdivisions residents. This is different than parks, playgrounds, and other amenities that the developer might build (aka "non-passive" recreation). In 2018, the county approved 156.52 acres of open space (about 8% of the total approved subdivision acreage) and $286,400 fee-in-lieu.