The search in the upper-right is the global search, not the task search.

But I still use it to search for tasks because:

1. it's RIGHT THERE, and
2. I naively think: "this'll probably work" before I proceed to search something completely generic like, "gerrit code" and it fails to root up anything useful.

fwiw, xlinking to the wild idea I shared at T397558#10938122 to redirect that top-right search-bar to the task search itself :p
(disclaimer: I have no idea about the feasibility of that idea, or whether it would be desirable by most people that use Phab, etc)

I don't think we should destroy the global quick search in the upper right corner (which would also be a larger downstream-only patch to maintain). I'd rather like to introduce a more limited default scope than now. See T397558.