What essentially everyone does care about, however, is repeatedly having rockets lobbed at them. When people feel under threat, reason starts to fall away, people begin dehumanizing the "other", and you get the massive mess we have today. The fact of the matter is that Israel will never accept any situation where its people are under threat.
I get what you mean, but the current situation has continued since even before the rocket attacks. Gaza was blockaded before rocket attacks even became a thing (setting aside the second Intifada because that's its own thing). What I mean is: Israeli's feeling under threat is probably a factor, but it's not the main issue.
and also, a legitimate Palestinian leadership in the first place
True enough, but let's remember that it's Israel that engineered a situation where they can claim Palestine has no legitimate leadership. You're not wrong about the fact, but I just wanted to make the cause clear.