I believe it’s correct to say that any one is usually (NOT always) followed by a prepositional phrase starting with of.
Anyone is a pronoun that takes the meaning of anybody or any person; any person at all.
Any one means any single person or thing; any single member of a group of persons or things; whatever one (person or thing) of a group.
Anyone could do this. = Anybody or any person could do this.
Any one of them could do this. = any single one of them could do this.
As to your final question, give us more background before we could definitively say yes or no. I probably agree with your take of it.