After extensive debugging, I've finally figured out the actual cause of the issue. Avast Antivirus.
Avast has a Ransomware Shield which "secures your personal photos, documents, and other files from being modified, deleted, or encrypted by ransomware attacks", and one of the directories in its paths to protect is the Desktop folder, for all users. Disabling this shield or removing the Desktop directory from it allows Forge to install to the desktop unimpeded.