A couple of comments, nothing new here.
Firstly, concerning the "rolling ball" effect and pincushion/barrel distortion (and I may note here that I do not own any binoculars that have barrel distortion), it seems that at least for most idividuals, their brains adjust to the type of distortion there is in the binocular they use regularly, and they cease to see the distortion and/or be bothered by it, sometimes even when they specifically try to see it. It is quite common when two birders try out each others' binoculars that both complain about exessive "rolling ball" in the others' binocular. The Zeiss Classic 10x40 was rather famous for having a lot of the "rolling ball" (we called it "carousell effect"), but many birders still use them happily and certainly don't consider it ever when viewing birds. Indeed, these types of distortions might well be more meaningfull to binoholics and collectors who use several different types of binoculars intermittently and do not give their brains time to solidly accommodate to any one image.
Secondly, all else being more or less equal, I personally prefer, like Henry does, binoculars that have the least amount of astigmatism in the largest possible area of the field. This is because when the eye/brain system perceives an image as unsharp, the eye will try to focus it, and when the unsharpness is caused by field curvature, the accommodation actually helps, at least partially. However, with astigmatism it doesn't help, as while the tangential focus might be improved, sagittal becomes more defocused and vice versa. My experience is that this tires my eyes significantly more than dealing with field curvature. This also relates to sweet spots. I am happier with a sweet spot which fades away into defocus than an equally large sweet spot which fades away into astigmatism.
I have compared two binoculars with field-flatteners, the Nikon SE 10x42 with a single-element flattener, and the Canon 10x42 L IS with a doublet flattener. The Nikon has a 60 degree apparent field, while the Canon has 65 degrees. Despite its wider field, the Canon has less astigmatism and field curvature towards the edges. Both binoculars have very large sweet spots, and allow my eyes to roam around the image without fatigue, but the Canon manages to maintain higher image quality further towards the edges.
So, like Henry, I can only see benefits from using field-flatteners, and based on the two examples quoted above, feel that a doublet can be better than a singlet. Back in the darker ages loss of transmission and contrast and increased scatter would strongly have argued in favor of simpler optical designs, but modern coating technologies seem to have changed that.
Kimmo