The point has been brought up earlier on the thread, but seems to have been marginalized: a "No-Mow" area and an ESA are two vastly different scenarios. Whereas an ESA is determined by outside agencies, a "No-Mow" area is a naturalized area of turf determined, usually with the superintendent consulting with the club pro, or vice versa, by finding areas that are most typically out of play. More often than not, the areas are in no way environmentally significant, but rather it becomes clear that not mowing certain areas will positively effect the maintenance budget (less fuel used for mowing and/or less labor). The fact that these naturalized areas often help contribute to environmental health, via habitat patches, filtration of water, etc., is simply a bonus.
They can (and perhaps should) be played through the green, and a lost ball is certainly a possibility. But, moreover, with varying skill levels of the millions of golfers worlwide, or even the hundreds who typically play your home club, we should all be able to admit that there is no such thing as an "Out-of-Play" part of a given golf course. Every inch of a property is in play. Surely people leave the wide fairways of Sand Hills, or Prairie Dunes every day? But, in that instance, the club cannot possibly mow down every area of the property where people might hit a ball, nor should they; it's too vast. The surrounding grasses are left there for budgetary and aesthetic reasons. A club in an urban context is simply trying to solve the same problem.