@evan Thanks for asking. I can say what jumped out for me: the implicit assumption that a Jewish person necessarily has any connection to Israel or to Israeli policy at all -- i.e., that a Jewish person participating in such a protest is more worthy of inquiry than any other non-Palestinian participating would be.
If the concern is just that the protests should feature Palestinian people more prominently, then that would argue against any non-Palestinian taking a leadership role -- not just against Jews doing so. (There is a complex question of what is meant by a "Palestinian" in this context, but for our purposes here I just mean someone with a personal, familial or other heritage-based connection to Palestinian territory and thus to the experience of oppression by Israel.) But your question wasn't "Should non-Palestinians have leading roles in anti-Gaza-war protests?", it was "Should Jewish people have leading roles in anti-Gaza-war protests?"
This is also different from the dynamic, which you offered as an example, of white people taking leadership roles at a Black Lives Matter protest. In that case, the dichotomy is inherent and unseverable from the political situation: at the very least, every white person in the U.S. has benefitted economically from years of pro-white-because-anti-black policies, and thus every white person truly is implicated, even if not by their own choice.
But a Jew outside Israel is not inherently implicated in Israeli policies, does not benefit from U.S. subsidy of Israeli military action, etc. They're just a person, who is Jewish, living in a country that is not Israel, as indeed most Jews have done for the past couple of millennia. Most importantly for this discussion, they have no way to further separate themselves from the connection that many people seem to assume they have with Israel and its government's policies. They're already doing all the same things everyone else is doing to achieve that! Like, you know, not living in Israel, not being a citizen of Israel, not voting in Israeli elections, etc. The unfortunate implication is that the one remaining thing they could do is not be Jewish.
That's why I asked "Why is this a question?"
@marymessall in reply mentioned that "Non-Jews don't need to be telling Jewish people what their roles are", and I want to be clear that that's not what I'm saying here. My reaction to your question would be the same whether you are Jewish or not (indeed I don't know whether you are), and my feeling about whether someone should be asked to stay out of a leadership role because of being a Jew would be unaffected by whether those doing the asking were Jewish or not as well.