Running anti-virus scanners immediately on uploaded files was a widespread practice in the BBS Era, adopted in response to the same kind of abuse. FTP sites did it, too. There was usually a delay for (run on a schedule) virus scanning between uploading to one of the shareware FTP sites and the file getting out of the upload directory and into the download areas.
The headlined article isn't really about hosting user-uploaded files, though. In no sensible system do user-uploaded files go straight into the site's own published JavaScript. The headlined situation is not analogous either to you or to BBSes, involving script hyperlinks to defunct third parties in one form of the problem, so what to do when it comes bad uploads from users is really going off on a tangent from Scott Helme's actual question.