I have only flown with a bike in the S&S backpack case. To me the convenience of being able to put my S&S case in the trunk of a taxi instead of trying to figure out how to get around with a full size bike box is just as important as the cost savings to avoid oversize package. I have even used the backpack straps to carry my S&S backpack as a backpack.
I would only carry a bike in a full size bike box if there was some reason that I could not use my S&S bike or my folding bike.
I also built up a DIY center support for that case, photo of my folding bike in that case with that support at this link. Built it with a couple wood dowels, small pieces of plywood and some wood screws:
https://www.bikeforums.net/20711320-post39.html
I worked in a bike shop years ago, I built up most of my bikes from the frame, thus I am very comfortable with disassembly and re-assembly. To me it is just lost time to do so. But some people are not mechanically inclined and I can see where they might not be very interested in that much mechanical work to pack or unpack it.