Facet - default value with search api
Not sure you can do this with facets.  Facets by definition show only options that exist in the current query results.  There is no guarantee that a content_page type will be among the search results for a term. 

You may want to consider using an exposed view filter (the views_block_filter_block module lets you turn this into a place-able block).  This would show all content types and can have a default type selected if the filter has none.