MyGov should support most desktop browsers, presuming the browser is mostly standards compliant (Chrome, Firefox, Edge, Safari, Chromium, etc.). Their new system seems to be a bit buggy, though, so perhaps that reflects in apparent browser compatibility.
Some info on the recent changes:
https://my.gov.au/en/about
https://www.servicesaustralia.gov.au...nged?context=1
MyGov has added a code generator app to the list of sign-in options:
https://my.gov.au/en/about/help/mygo...gn-in-to-mygov
I'm a tad disappointed, that you can't "fall-back" on the fly to a second sign-in option if one fails, e.g. if you've lost your phone. What it looks like, at the moment, is that the only recovery option is to use myGovID, which is complicated and probably also accessed via an app on your phone.