And it's my day to try to help you
Your png icon is a good method. But it will actually not appear on the game file which will unfortunately keep its original exe icon

If anyone has succeeded, please tell us. Even if you try to change icon yourself by right-clicking on game file, there is no option present that will allow you to do so.
Where your icon is visible is when you run the game. On the left top where the exe icon was previously present, you'll find your icon
Also whenever you have two games in one (ie first game loads second game) using 2 different icons, in the 2 different games will be found 2 different icons.
One other thing

.ico files (icon files) don't seem to work in GE. jpeg works well but unfortunately won't have alpha.
There ! I think that is all I have observed from icons. Hope it has helped you, friend
