There are multiple Flash Player plug-ins. The PPAPI one for Chrome, the NPAPI one for Firefox, and the ActiveX plug-in for Internet Explorer. If I had to guess, Steam and Origin are probably using an embedded instance of Internet Explorer. I would recommend that you perform the installation from Internet Explorer by going to http://get.adobe.com/flashplayer, or at least confirm that it's installed from here: Flash Player Help
If Flash Player in Internet Explorer is installed, I'd check with the Steam and Origin support forums to see if they can offer any insight.
Good luck!