Israel has pretty strong left wing and extreme left wing, as manifested by Israelis support for parties like Hadash, or organizations such as B'Tzelem, Breaking the silence and so on. The extreme left may particularly support one-state solution (popular among European and American left, but contrary to the two-state solution supported internationally by most western and even Arab states.)
This also can be seen from reading mainstream Israeli media, such as Haaretz (the Israeli equivalent of New York Times/Guardian/LeMonde, available in Hebrew and English.)
However, one should not confuse criticizing Israeli government and policies with criticizing "Israel" itself: even the leftiest Israelis do expect that in the end they will live in peace and security alongside Palestinians - none of them is planning to "move back to Poland" or elsewhere, as some Israeli "critics" occasionally suggest. Particularly not those who were born in Israel, especially in second or third generation.
There are also many prominent Israelis, critical of the government, but nevertheless supportive of their country - like Natalie Portman.
One thing to keep in mind is that only 73% of Israelis are of Jewish origin - the rest are mostly Palestinian Arabs and other ethnic groups, such as Druze, Arameans, Armenians, etc. Palestinian Israelis are often critical of their country, as can be seen by their representation in the Parliament, which until recently consistently refused to join any government coalitions. Some of the Arab politician voice publicly the views as extreme as those of BDS like Haneen Zoabi.
In addition to Palestinian parties, there are also mixed Arab-Israeli party Hadash. There are also Palestinian members of what is traditionally considered Jewish parties - like Ghaida Rinawie Zoabi, who is a member of Meretz, the party of Jewish intellectuals.
The rest of the political spectrum is split among more mainstream left, right, and religious parties (unlike in the US, religious and right-wing views are not necessarily linked - in the past religious parties also took part in left-wing governing coalitions.)
The full breakdown of vote for various parties can be found on Wikipedia: 2022 Israeli legislative election.
It is necessary to note that political participation nearly excludes adopting BDS positions (with some exceptions in the Palestinian Parties and Hadash.) Indeed, Zionism is Jewish movement for national independence, and participating in the political system of the Jewish state means being a Zionist. In this sense opposition to Zionism is opposition to the existence of the Jewish state, a member of the United Nations, recognized by significant number of countries, including all of the western world.
In this sense, the use of terms Zionism and anti-Zionist by BDS, anti-Israeli groups, and some media coverage is misleading, perhaps deliberately so, and the OP seems to be influenced by this use. It is also worth stressing that Zionism from its inception was a secular movement for the creation of the Jewish state, as opposed to the religious imperative to move to the lands granted to Jewish people by the Almighty - in this sense religious Zionism is a misnomer.
Zionism and religion
Naturei Karta is a good example of how Zionism is in contradiction with the religious thought, since it implies that the laws of the state of Israel supersede the Laws of Moses. In a broader sense, there are significant tensions between Israeli governments and the Jewish orthodox religious communities, regarding the military draft of yeshiva students, obligatory education program, interference by social services in child abuse, etc.
This also applies to the settlement activity: evacuating West Bank settlements built without appropriate permissions is also not an uncommon practice (e.g., those outside of the areas designated for building by the Israeli government or on the land not owned by the settlers.)
While Zionism is sometimes used in a very broad sense - to designate any activities associated with Israel and/or Jews, it is largely agreed that anti-Zionism implies opposition to the existence of the Jewish state:
Anti-Zionism is opposition to Zionism. Although anti-Zionism is a heterogeneous phenomenon, all its proponents agree that the creation of the modern State of Israel, and the movement to create a sovereign Jewish state in the region of Palestine – the biblical Land of Israel – was flawed or unjust in some way.
Note that this means opposition to the Jewish right for self-determination, as defined in the UN charter
All peoples have the right of self-determination. By virtue of that right they freely determine their political status and freely pursue their economic, social and cultural development.
Interestingly, some forms of anti-Zionism, notably those favoring one-state solution, also deny self-determination to the Palestinian people (i.e., neither Jewish nor Palestinian state.)
Here is a short selection of articles published by mainstream left-wing daily Haaretz since October 7. The content is behind the paywall, but the titles speak for themselves. One can also read the comments section.
October 8: Editorial | Netanyahu Bears Responsibility for This Israel-Gaza War
October 19 Opinion | The War in Gaza Must Stop Immediately
October 21: Opinion | The Secret to Netanyahu's Gaza Failure: Do Nothing
October 23: Opinion | In Gaza, Israel Is Racing to the Moral Abyss
October 26: Opinion | It Is Forbidden to Even Empathize With Innocent Gazans
December 10 Analysis | Netanyahu Must Be Politically Destroyed, or Israel Will Go Down With Him
Here is from the right-wing The Times of Israel:
October 8: Op-ed | For years, Netanyahu propped up Hamas. Now it’s blown up in our faces
I think these qualify as "profound criticism of Israel". Their main distinction with the rhetoric of Israel's critics from abroad is that, while supporting Palestinians and criticizing Israel, Haaretz writers are not anti-Israel and not anti-Zionist.