Wow, interesting question. For sure. And, actually, it is one that I have thought about, albeit decades ago, not recently.
There ARE negative connotations to the word "Jew". There are echoes and ghosts of lingering anti-semitism. Some elements in the larger society attaches negative judgment to being a Jew, which for them is the "other". That has had terrible consequences, as you must know. And, honestly, I don't think that is entirely over yet.
Nonetheless, I am a Jew. If anyone has a problem with that, it's their problem -- not mine. If some parts of society put some negativity onto that identity, me using some other form of labeling is certainly not going to solve that problem. One way to counter that negativity is to apologetically claim the identity.
When I grew up -- and even today -- there were organizations called things like the Young Mens Hebrew Association and the Baltimore Hebrew Congregation, instead of the Young Mens Jewish Association and the Baltimore Jewish Congregation. I think that choice of public naming was a defensive measure by Jews who found themselves needing to negotiate life in a society dominated by a culture which imposed negative judgments on Jews. Possibly they thought that Hebrew had fewer negative connotations than "Jewish".
Anyway, Fan, that's a good question. Thank you. I hope my reply offers you some understanding of my own personal perspective.