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When copying a file, instead of showing a progress bar would it be simpler to use the notification timeout animation in the corner which can also be used as a minimise requester button at the same time:

That would mix two separate things.
A timeout is a timeout, it is expected to be precise and not to skip or hesitate.
The progress of a file system operation is different, its is influenced by various things.

To me this mix would be confusing in exactly those moments when it comes into focus.

You have a point, I was more intrugued by the look of a circular progress bar