Almost nothing has changed on that page with the introduction of the paste button (clicking the upload button still calls the exact same code path), so I wonder if this is an issue with the dropzone which accepts files. What version of Chrome are you two using, and if you drag and drop a file over the SGF library list does it upload as expected?
One way to test if this is the root cause might be to go to this page and try clicking the “Open File Dialog” button. If it doesn’t work, then this dropzone bug is almost surely the cause.