From experience, wheel size has the dominant impact. Remember the Birdy has ERTRO 355mm wheels so there are no knobblies available for it. A wider tyre which allows softer pressure also helps a lot. But I suffered similar problems on the Mini with 50mm Big Apples, although they were far less skittish due to width and lower pressure. But they also sank into softer surfaces.
The Birdy needs some really fat ones for off-roading, but softer stuff demands bigger diameter. I have been toying with the idea of a frankenBirdy with 406 wheels on there.