Just to add my voice, I’m having the same problem, too.
The problem only seems to affect post types added manually using register_post_type
in functions.php, though. I didn’t have the same problem with post types added using the ‘Custom Post Type UI’ plugin on another site, so perhaps it’s worth trying this (or another) plugin if you are having problems.
This makes me wonder if there is some property that is being set by the plugin that needs to be set in the register_post_type
function for the Media Picker to work.