How to make a variable "public" and editable when the eye icon next to it does nothing when i'm clicking on it ?

So the same thing happened to me in UE5.1

In UE5.0 you can make the variable for any widget element and can manipulate in the event graph
In UE5.1 and onwards you cannot make the changes by making it variable rather what you can do is you can create a variable and bind the widget element to that variable

This works for me, hope so this also works for you guys!