Thanks for pointing out your frustrations with WordPress' HTML editor. What you are experiencing is completely accurate, and is actually considered a "security" feature from the WordPress development team's perspective.
TRUEedit currently only lets you configure the frontend content filters, such as wpautop. The filter that removes most HTML tags from non super admin (non admins in single site wp) is not a frontend filter.
Also you should keep in mind that even if I fixed your issue with the editor stripping out HTML code, the XML export and other features of WordPress will continue to strip out your custom HTML code. Basically, if you want someone to write full blown HTML they need to be an Admin.
With past clients I have used other plugins, such as Viper Video instead of copy and pasting HTML embed tags for YouTube videos. Perhaps you may rework your CSS so the preformatted tags do not a specific class? In other words, you might try looking for other way to implement what you need.
I can feel your frustration, and when I get some time I will look into adding a feature into TRUEedit that will allow for full HTML editing capabilities for specific users. It might be edits to numerous different hooks and actions, so I am unsure what the scope of a feature like that will be.
Cheers and thanks for your interest in TRUEedit.