If I were in charge of NASA I would make their sole mission to find any kind of life in space with proof. Such a find would be bigger than landing a man on the Moon as we did 40 years ago and as you say leave manned space flight moving forward to private industry.
That's too ill-defined a goal. Where do you look? What if there's nothing to find?
A better goal for NASA would be to establish a permanent human presence beyond LEO
by X date. Maybe a moon base, maybe a Lagrange station, but something that takes our current achievements and pushes them to the next level.