Of course people care primarily about the actions of their own democratically elected government, that's the whole point. There's no need to protest when people agree with their government.
Agreed any real solutions are a nonstarter in current situation, but a lack of imagination or will about how to move forward just further normalizes the illegality of it all.
It could do a lot in the West Bank (where the fully or partially PA administered territory is divided into 166 non-contiguous regions), and anything there xould be done in a way that it looks like a win for the Fatah-led PA, weakening the perception that Hamas and its violence is the only entity capable of delivering for the Palestinian people, undermining Hamas politically.
OTOH, the whole reason Israel fostered Hamas during the direct occupation of Gaza was to create an Islamist competitor for the more secular and sympathetic to non-Muslim states PLO, and the reason they've (and government ministers have said this explicitly) continued to support them in between periods of active conflict is to deflect pressure for peace and a two-state solution, so there’s zero chance of the Netanyahu government doing this.
Not almost. The far right Israel factions (Netanyahu, Likud, etc.) have actually repeatedly encouraged and cultivated Hamas. They benefit far more from the polarization that Hamas brings than a mild and moderate PA who is willing to work diplomatically, because then that increases pressure on those far right Israelis to also be more temperate, which goes against their goals.
Funny since Israel originally supported (including arming) Hamas* hoping the religious Hamas would split the populations support for more secular nationalist movements in Palestine. But, you can be both religious and nationalist.
*Hamas/Muslim Brotherhood; Hamas grew out of the Muslim Brotherhood.
Israel and especially its present governing coalition is not blameless for the situation (and nor are Palestinian factions and some of their supposed allies blameless for Israel's tendency to keep electing governing coalitions more interested in projecting power than continuing peace processes), but it's a lot more complicated than Israeli govts wanting Hamas to be a thing and nobody else in the region having agency. Undoing tacit support for an Islamist alternative to the PLO in the 1970s isn't really a policy option (if it is, someone should give the undo button to the US for Afghanistan!), that happened because there was open conflict long before Hamas and Netanyahu, and apparent diplomatic wins for the PLO did them absolutely no good in the noughties when Palestinians could still choose whether or not to vote for therm
Yet there's large protests in countries that aren't allied with the US or Israel, when no such protests were forthcoming in other analogous scenarios. And there were scarce protests in the US against Saudi Arabia's campaign in Yemen despite US alliance.
I do agree that your thesis is a partial explanation, but it is far from a full explanation. There's two other things going on.
Among Western leftists and minority groups, Israel is a symbol. It's perceived as the last vestige of Western/White colonialism. A symbol of someone with white skin punching down on brown skinned people. It harkens back to the reason that your ancestor was forced (either literally or by material circumstance) into the US in the first place, and why you are living today under systemic racism. Defeating this placeholder is therefore an important milestone in restoring their sense of historical justice. Needless to say this is oversimplified given how many Israeli Jews are indigenous to I/P or were ethnically cleansed from the surrounding MENA area and forced into the I/P area, but people do legitimately hold that dichotomous oppressor/oppressed worldview.
Among Muslim countries, this is an ethnoreligious blood feud. Assad killing Muslims doesn't cause the same anger because it's within the same identity group. So it's a classic case of identity divisions leading to disparate anger. I'm massively oversimplifying here, there are many other factors, but it's part of what's behind the energy.