Is my cash safer with a big bank?
Not necessarily. All of Australia's Big Four are Authorised Deposit-Taking Institution (ADI). But so are almost all the smaller banks, credit unions and digital banks. You can find more detailed information here.
In the unlikely event your lender goes bankrupt, it won't affect your ownership of the property as your loan account would pass to a new institution, who bought the assets of your now-defunct lender.
The only lenders who might not be covered by the bank guarantee scheme are very small neobanks, which are essentially financial start-ups. Most of these companies obtain a banking license or partner with someone who has one, and is an ADI. .