so does pretty much every private residence.
Exactly. Those who argue against discrimination here have a much better distinction with just the "public accommodation" or "generally open to the public" distinction. That distinction is pretty clear and while there are some gray areas, the vast majority of property can easily be separated into the two classes by anyone applying that test in a reasonable fashion.
BTW, the distinction between government and private property has gray areas too, e.g., businesses that are largely dependent on public funds, most colleges, hospitals, etc.. Then there are the quasi-private entities, e.g., the USPS, Amtrak, etc.
There are almost no tests or classification methods that always work in distinguishing between different types of things. For instance, plant vs animal. Seems pretty clear for a cow vs a fern. But then that damn sponge, coral reef, Venus fly trap and other things come along that leave one a little puzzled. I have gotten into arguments with people that did not think insects were animals, though it's clear they are.
The public vs private road and access to electricity, water, sewage, etc. does not work at all. I can came up with easy examples of businesses that could hide behind the private road and, as noted, that does not work for separating private residences.
