With just white walls, the higher On/Off projector will sometimes look better than just a brighter projector that has poor On/Off contrast, that is true depending on just how close the walls are and how reflective they are back at the screen.
This concept is misunderstood on both sides, I studied it from whitepapers, let me see if I can clarify why this confusion exists...
If a room is bad enough, as the ANSI contrast is severely lost, our eyes start seeing the details blend together. Even if technically the white peaks and black floor still has the same contrast distance (On/Off), it won't matter because the intrascene contrast will be lacking detail due to the loss in ANSI within the scene and all details will blend and shadow detail lost even in the darker scenes. Native On/Off is still a white to black measurement and not an intrascene measurement, and it may be a great indicator in a normal room of how good a projector will do frame-to-frame contrast in dark scenes, but eventually there is a point of no return where too much loss in ANSI contrast messes up the entire picture, including On/Off scenes and scenes needing ANSI.
At some point, a projector displaying a higher On/Off contrast but with NO ANSI due to a terrible room would basically look like a flashlight turning on and off, it would just be a plain white screen or a plain black, there would be no detail in the image. Our eyes find it harder to differentiate details in dark scenes sometimes, so if you start losing intrascene contrast in dark scenes, you won't really care how dark or bright the image is, it'll just look so flat and muddy it won't matter.
We don't think of ANSI affecting darker scenes because projectors already have high enough ANSI for dark scenes, and ANSI isn't a big indicator of intrascene contrast in dark scenes in most rooms where projectors are used simply because they all have enough ANSI in a good room, but there is a catch here. If you make the room bad enough and lose too much ANSI contrast, now all of a sudden even darker shades of gray in dark scenes are not gradiated in the intrascene contrast because there was just too much ANSI lost and ALL details are blending together in bright and dark scenes. The three dimensional look we obtain in darker scenes is still due to both ANSI and On/Off, but you only need a tiny bit of ANSI and a whole lot of On/Off contrast, so we generally ignore the effects of ANSI and intrascene contrast in darker scenes. The problem with ignoring it in a poor room is eventually you lose that tiny bit of ANSI you needed, and who cares anymore about the On/Off.
That is the hole in the theory, even though ANSI contrast doesn't matter much in dark scenes, it eventually will if you lose too much of it.