There are a few questions here, which are all tied up in one another. First of all Israelis are in general not afraid to be critical of the government and institutions and since the beginning of the war there have been voices criticising it (including from members of the kibbutzim that were destroyed on 7 Oct). Gideon Levy (who is a well known leftist journalist in Haaretz) is hardly the only one. People are often critical of him but more on a personal level because of the tone of what he says, including making out that he is some kind of lonely persecuted voice. There are small groups of people who have demonstrated against the war throughout.
Having said that, I have only recently seen more leftists starting to be more critical of the war in general. This is not because they are somehow censored from saying it, but because most of us are genuinely extremely conflicted (and also know that whatever we say on Facebook or to friends won't have an impact in any case). This is relevant to the other part of your question - 'what is the uk not getting that Israel does?'. The difference is not so much a difference of information, it's a difference of perspective. For someone in the UK, thinking or writing 'ceasefire now' is probably easy to say because you don't directly feel the consequences of what happens here. For us here in Israel, the state in which Hamas is left following the war has a very direct and immediate impact on the safety of our friends and family who live in the south and Gaza border areas. Even though I am both critical and extremely sad about the way the war is playing out, and have extremely deep criticisms at many levels of the government and its motivations, before I would write 'ceasefire now', I also need to be reasonably convinced that my friends on a kibbutz 3km outside Gaza, which Hamas intended to attack on 7 Oct but didn't succeed, will be able to return to their home and sleep at night in relative security even knowing that Hamas has openly declared its intention to attack again. Likewise, if I say that I'm fine with releasing all Palestinian prisoners in Israel in exchange for the hostages, I have to bear in mind that among my friends there are both family members of hostages still held by Hamas, and also people who have been bereaved in terror attacks masterminded (like the 7 Oct attacks) by prisoners released in a previous hostage exchange. For all these reasons, for most Israelis it's complicated.
Are we sheltered? In terms of news images, as I've discussed previously, we see fewer violent images (from both sides) than TV news overseas will broadcast. The Israeli news consciously tries to limit the violent images, not because they are trying to censor the existence of events, but because news is quite often on in front of kids. In terms of the impact on Gaza, it's reported in pictures, numbers etc but there wouldn't be, say, a news reporter on the ground reporting events from a Palestinian perspective.
However, I could hardly describe any of us here as sheltered - again, the perspective here is completely different. We are all affected directly by the war, all the time. We all know multiple people in the army and multiple bereaved people, and multiple people with hostages in their families. Our world has been turned upside down by what happened on Oct 7th both directly and in terms of the way it has made us reassess what we believed about our own safety. I guess as wise people have often said: don't judge people until you have seen the world through their eyes.