That's an interesting idea, but standardized names for the characters would be necessary. "Star" is colloquial; "asterisk" is more appropriate. Instead of "minus," "hyphen" seems more correct and what about the unary versus binary overloads? Here's a longer list to consider:
has_operator_hyphen_unary has_operator_hyphen_binary has_operator_tilde has_operator_exclamation has_operator_percent has_operator_caret (not "circumflex" which is a diacritical) has_operator_ampersand has_operator_asterisk has_operator_pipe has_operator_left_guillemet? has_operator_right_guillemet?
Since the guillemets are real characters, and the C++ operators are formed by two characters parsed as one token, using "guillemet" isn't correct. One could also use "chevron" which are sometimes doubled to represent guillemets, but the less-than and greater-than characters are not chevrons. Calling them "left shift" and "right shift" brings one full circle to naming the operators for the operations they represent on built-in types, while "insertion" and "extraction" are related to their use with IOStreams. I don't know what to call those.
Oh gosh, please no....! Lets use names relating to what the operator does, not what obscure names we can think up for the punctuation. John.