The file associations work for me (and have always worked) on Win 8.1 with the disc version of the game. I'm attaching a screenshot that shows the Registry key (presumably) responsible for loading savegames from Explorer. Well, I don't suppose that the BtS Registry keys just somehow went missing on lymond's and Nexus's systems. Seems likelier that the system expects them in a different place or that Firaxis's fxsload argument no longer works. One could test the latter hypothesis by running Civ4BeyondSword.exe with that argument from the command line, as in the second attached screenshot. I had copied the QuickSave file into the same folder as the EXE for this, and only the fourth attempt (with slash, equals sign and quotation marks) worked.
And
this post in the C2C subforum describes how the associations can be set up without editing the Registry – but I'm not sure if anyone has successfully tested that.
I don't think either approach (regedit, cmd) can work for the Steam version.