The problem with the extension is that it assumes if no price is given on the site, it is free and should show the download button. However, the Play Store Markup changed to always display the tag and use zero for free apps instead of no price at all.
Just change line 17 in google_play.js in the extensions js folder to:
showButton = !price || price.getAttribute("content") == 0;
After the change, disable and reenable the extension for Chrome to catch up on the change.
On Mac OS X the extensions are stored in ~/Library/Application Support/Google/Chrome/Default/Extensions/. It's one extension per folder with rather cryptic folder names. The APK downloader is one of those.
On windows google_play.js file is at
%USERPROFILE%/AppData/Local/Google/Chrome/User Data/Default/Extensions/jdpglblnnaocjhfenhockgamhoogihfi/2.0.0_0/assets/js