I've read and watched so much over the last month, that I believe that it's a mixture.
Some Israelis will know and support it as a means to an ends (e.g. No 2 state solution, settlements cam be built).
Some geopolitical commentators I watch have actually said that the Israeli media is quite open in things like stating that how the Palestinians are treated is an Apartheid in the media, so the public should be informed (Haeretz is brilliant and has reported on the Bibi Hamas connection)
However the IDF is not above using psychological operations on Israeli citizens to manufacture their support:
The IDF has admitted it engaged in “psychological warfare” during its 2021 military operation in Gaza by using fake social media accounts to try to increase public support for its operations in the Strip.
The IDF’s Spokesperson’s Unit, which serves as the army’s media arm tasked with coordinating domestic and international media campaigns, was found to have opened and operated a range of fake social media accounts in the opening days of the IDF’s Guardian of the Walls operation, according to a expose published Wednesday in the Haaretz newspaper.
The fake accounts were reportedly designed to foster greater positivity among Israelis towards the IDF’s actions in the Gaza Strip in response to a perceived lack of public enthusiasm...."
https://www.timesofisrael.com/idf-admits-psychological-warfare-attempt-against-israeli-citizens/
So if similar techniques to shape opinion are used then many will write it off as a Conspiracy.