Flash 10.2 and newer has a 'feature' that will not play SWF files that are delivered with the "Content-Disposition" HTTP header of "attachment".

You can get Kentico to send SWF files as "inline" disposition with the following change:

~/App_Code/CMSPages/GetFilePage.cs:277

if (ImageHelper.IsImage(extension) || ImageHelper.IsHtml(extension) || extension.ToLower() == ".pdf" || extension.ToLower() == ".swf")

which just adds the ".swf" extension to the condition.

Thanks to Kentico support for pointing me to this.