Well yeah. That's not really the point, though. Aside from existing code, hard-coding is generally frowned upon. Our devs have been using CURRENT_DATE and its ilk for over six years now.
Would it help to put the current_date call in a wrapper function and coerce it as IMMUTABLE? A quick test shows that constraint exclusion seems to kick in, but I can't speak intelligently about whether that is wise or not.
Or what about something like DATE_TRUNC("DAY", now())? Or would that run into the same optimization/planner problems as CURRENT_DATE?