>Central Park worked because the existing density was already high and there was no way for Manhattan to grow outward

Central Park was built before the first subway and before the bridges connecting Brooklyn, Queens and Jersey. The dirt dug from the tunnels was added to lower Manhattan. Prior to elevators buildings rarely went higher than 5 stories. There was in fact ways for Manhattan to grow outwards (effectively).

Hmm, but consider that it works in both directions. In places that are not as dense and expensive, the public places won't increase rents of the surrounding properties as much, but neither will they take away as much tax revenue. Both scale proportionately. I think this applies just as much to a park in small town as it does to Manhattan.

most of that population was still well south of the Central Park area. Streets had been laid out, and the occasional building erected in the area. But the population density was downtown, in the tenements.

Astor and others benefited as population expanded north.