The only thing I can think of would be a javascript conflict between some script, or plugin, and the platform. I have tried everything on my test site with all four of my themes, and the other base themes as well a couple of others. I moved widgets around, changed icons, modified the welcome message. Everything is working. The functionality, and elements of the customize page pull from the core platform. The styling of the elements will pull from the css in the theme.
Other than a conflict between the core, and any additional script added by the user, or 3rd party plugin, I have also seen that if one has Cloudflare, and Rocket Loader enabled that this will also interfere with the core platform's javascript libraries. I don't have Cloudflare running at all.