Bypass the Unwelcome Screen by creating Desktop shortcuts for the Organizer and the Editor, and restarting your computer.
The Editor's executable is:
"C:\Program Files (x86)\Adobe\Photoshop Elements 10\PhotoshopElementsEditor.exe"
The Organizer's executable is:
"C:\Program Files (x86)\Adobe\Elements 10 Organizer\PhotoshopElementsOrganizer.exe"
The Unwelcome Screen leaves a zombie process running which prevents the actual program from launching again.