I have both "step in" and "strap on", for various reasons. However, most modern crampons are a hybrid of both designs.
I use my step-ins with plastic boots, ski boots and rigid mountaineering boots. All of my step-in binding are "hybrid" in that if they have a bail on the front, they also have a strap that goes from the front to the back.
(see the metal strap going from the bail to the straps at the back).
This means that if the bail pops off, there is still a chance that the crampon can stay on your foot. The first rule of crampons is that you have to keep them on, or carry a spare!
This design is flexible under foot, but if the boot is stiff, it will perform like a rigid crampon.
For the SAR team we buy the ones that are in the video, which are similar to this:
This design works with most boots, is instantly resizable, and is pretty idiot proof. We carry them in the equipment truck, and issue them to members sometimes. The horizontal front points are good in stiff snow, where the vertical points might cut through, but they still work very well in steep waterfall and glacier ice.