That's another issue easily solved by DnD API modern browsers implement. And should probably be done differently with regards to UX depending on whether the clicky part was a button, or a larger upload area.

Frankly, I don't find the original UX of dragging files onto an upload button all that intuitive. So instead I add drop functionality to a larger container in my apps. So if I have dialog that accepts files, users can drag files anywhere over the dialog and the dialog highlights that it accepts drops while dragging.