Am I the only one old enough to believe to remember that IE had this functionality (I refer to the "full screen on click on something in the page" functionality) at least more than 10 years ago and it sucked big time, as poor users would just click somewhere and then they would end on some malicious page which would "replace" their menus, taksbars etc with some nefarious code?

Why should it be different this time?

Because mozilla has given it some thought https://wiki.mozilla.org/Gecko:FullScreenAPI

How does this compare to Chrome's kiosk mode?