"Might be basilisk-y" - especially as written by a human who is clearly communicating intelligibly and is not compelled to spread the basilisk to one and all - does not indicate a great deal of danger. I'm pretty psychologically resilient; I know the other one, which I find to be only a vague, occasional background distress (though that one has done some people real harm, and I don't advocate spreading it around, I haven't found it a serious ding to my quality of life personally). Axel has some non-zero ability to identify basilisks and wanted a second opinion; if asking for a second opinion on potential basilisks never resulted in volunteers I think this would probably result in more publicly revealed unchecked basilisks rather than fewer. (Not sharing an idea you have with anyone is hard.)

(I don't think Axel's idea is particularly basilisk-y, especially not in a context like this one, and have responded thusly to his PM.)