If really all meta data and content of all searchable objects should be searched, the default search would be of worse performance than most of the advanced search requests.
That can be discussed. Perhaps the default search would only search metadata? That would be relatively fast, and the user could then select "metadata and all content" in the advanced search.
That would not be the expected behaviour. A user would expect that a search located in the top frame would be a global search.
I agree, the expected behaviour would be for it to be a global text search. But you said (correctly) that it would be slower, so I offered a suggestion to make it faster. Either it works globally, and slower, or it searches a smaller subset of the data, and is faster. We may choose one or the other. For myself, I would prefer the gobal, but slower, search. Which would you prefer? I will work to implement whichever the ILIAS team prefers.
If we want to provide a search in the top frame it should at least include an object type selection list (learning module, forum, ...) or a similar filter...
That is a bad idea, for two reasons:
1) It will require more space in the title area. As you have pointed out, functions implemented in the title area should use space efficiently, because there is a small amount of space to work with.
2) It's clunky and adds complexity, which is in opposition to the entire idea of having a simple, one-click Search function. Filters and other added details are suitable to the the "advanced search", and that is how users will expect it to work.
I will also read the document you linked to.