How to give a roll only access to add a certain content type?
The Permission View published content is very important, without the users of that role won't be able to see Published content. What do you mean by "cannot do anything in the system anymore"? 

Do you want that Role to be able to create new nodes of that type, but not to be able to View those nodes once they're created? 

You should look into the Editorial Workflow, and Content Access module.