<<<If one such aircraft does exist, why not leave the flaps out permanently.>>>
This trick was tried by a TWA 727 crew some years ago, at high altitude, with rather poor results...a bent aeroplane, and a wild ride on the way down.
I did not operate earlier models of the Lockheed Constellation, but the 1649 had very high lift fowler flaps, and improved all engines Vx climb rate could be achieved with the lowest amount of flap extension (flaps 5, I think), at the same speed (V2+30) as with flaps up.
The 1649 had a newly designed wing, which was very efficient.
A very nice aeroplane.