By Dom Nozzi
I have a view that seems counterintuitive to many, and therefore tends to draw indignant ire from many intelligent, well-meaning environmentalists who seek to create a better city.
My seemingly shocking view is that introducing nature into a town center nearly always degrades the town center. I say this even though I have a degree in environmental science, which gives me a thorough understanding of the importance of protecting and restoring natural ecosystems.
I also say this even though in childhood, one of the most profound, critically important, priceless experiences I had was to be able to play in the neighborhood woodlands. I did that all the time. In fact, the main inspiration for my becoming a city planner was that I wanted to be in a job in which I could work to see that future generations of kids had that same opportunity. My childhood experience with nature made me realize that not having that exposure to nature would lead to an awful, sterile, barren childhood. Indeed, a research study once evaluated a large number of variables to determine if there was a correlation between childhood experiences and wanting to conserve the environment as an adult. The study found that there was one variable that stood out head and shoulders above the others. Adult conservationists typically were able to engage in unstructured, unsupervised play in natural areas near their home when they were kids.
For the above reasons, as well as a strong interest in promoting transportation choices in cities, I am a leading advocate for establishing urban greenway trail systems in cities. Greenways provide an effective way to allow kids (and adults) to have easy walking/bicycling access to the natural world, on a regular basis, right outside their back door. I know of nothing that is better able to create the army of conservationists than greenways.
However, I must add an important qualifier on the topic of introducing nature to a town center. The urban town center habitat (in contrast to the suburban and rural locations of a community) must be compact and walkable if it is to be a high quality urban habitat. That means that if we are to introduce nature into the urban world, we must be as careful as if we were planning to introduce human activity into a sensitive wildlife habitat.
In the case of the town center human habitat, the introduced nature must, again, be compact and walkable. In other words, small, vacant woodlots, plazas, squares, piazzas, utility corridors, creek corridors, and other similar, relatively small spaces are perfectly compatible with walkability. One can 
easily walk from origin A to destination B without an enormous amount of physical exertion. By contrast, putting a golf course or even a 50-acre park in the middle of a city creates an utterly unwalkable condition, as the distance between Point A and Point B then becomes too excessive to easily walk.
Central Park in New York City is an exception that can work because that city has extremely high densities and a quality transit system that means you can easily walk or ride to all of your daily needs along the perimeter of the park without having to cross it on foot. In cities with much smaller densities (and nearly all American cities have quite low densities compared to New York City), big open spaces in the town center would create unwalkable spaces that would degrade the urban habitat.
The key for nearly all American cities, then, is to preserve and create compact urban open spaces while retaining walkability. Greenway trails that wind their way through neighborhoods, as well as small parks and squares, are compatible.
Big, unwalkable parks are not.
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Dom–Most major cities have one (or a few) very large major park(s). Central Park in New York, Philadelphia’s Fairmount Park, Boston’s Emerald Necklace, Baltimore’s Druid Hill Park, Pittsburgh’s Schenley Park, Chicago’s Millennium Park, Frisco’s Golden Gate Park, London’s Hyde Park, Paris’ Bois de Boulogne, Berlin’s Tiergarten, and so on.
While I agree that the town center is no place for nature (except in the highly simplified form of a neighborhood square), I would also like to point out that these large urban parks have a different role to play, and are never of the city center; in a few circumstances, they border it, but they are another organism entirely.
Briefly, what little nature can fit in the town center needs to be limited to a communal lawn and play space; Central Park is a bad example for your post because it fills an entirely different role. Manhattan just happened to grow and densify around it, but Central Park provides the service of offering an respite from the city, of quietness and hiking and biking trails and woodlots and waterways. It is not like Bryant Park or Union Square in that regard, no more than Hyde Park is like Manchester Square or Hanover Square.
(That is not to say that all cities should be able to provide such facilities; they exist, for good reason, only in cities so large that access to the countryside has ceased to be convenient for much of their population. However, not all large cities that fit that criteria have developed such parks: Detroit is an example of the latter, as is Los Angeles.)