I also understand the appreciable filtering behaviour of the search field. Some time ago, I proposed an adaptive behaviour: show the filtered results, then collapsable the global results. When there are no filtered results, the global results are decollapsed, but if there is a filtered result, the global results remain collapsed.
At any rate, another solution might be Spotlight to search also amongst tasks (actually it only seems to search through projects, right?).
One last suggestion: I often want to do this global search to make sure not to doubly add a task. We could also imagine a live, global search through existing tasks, while entering a new task description.