Are {} "operators"? I don't think of them as such.

To me, most of the aforementioned uses are conceptually the same. Braces surround something and return its result. Syntactically there is some difference, eg if ${foo}_bar really acted like a block then the foo would be a bare word, but conceptually if you think of braces as a block which is strongly bound to whatever is to the left of the opening brace, I don't think you'll go far wrong.