In a certain sense, a mall is a public space, though. Meaning, you can't exclude people simply because they belong to a protected class. A privately-owned mall is very different from a privately-owned home or club in that regard.

I don't see why you could exclude people from a private club due to their protected class either. Homes are not businesses. This isn't convincing.

Are you disputing whether they should be allowed according to our current laws, or whether they actually exist? There are plenty of women-only or men-only or black-only or Jewish-only or Chinese-only or ... etc. private clubs/organizations. For example Augusta National Golf Club only recently admitted women, and solely due to public pressure / protest; not due to any legal requirement as far as I know.

Whether they should. I'm absolutely sure there exist organizations that break the law exist (sidestepping the issue of whether or not this does break the law).

Actually, some US states (including California, Massachusetts and Colorado) have concluded that malls are equivalent to public "Main Street" space and must be open venues for public speech. This notion is disputed elsewhere. See https://www.mprnews.org/story/2015/12/22/protests-first-amen...

Too bad malls are on the decline.