There is an easier way, that does not require modifying the plugin.
When you upload using the plugin's 'upload new profile picture' button, the image *does indeed* get uploaded to the Media Library, even though there is no success message and the open dialog is blank and stay open - simply refresh the (user profile) page, then click on the 'upload new profile picture' button again, but this time use the link/tab that says "media library" instead of uploading a file. Choose the image and then on the "show" link - be sure that the image attachment URL is in the box where it should be - then click on the button that says "make this my author profile image".
I've used this method now on several current installs of WP (currently 4.1.1) and it works fine.