It's tubs that shrink with inflation, although this alone isn't enough to stop them rolling off and they still need glue, shellac or double-sided tape for security. It's true that wired-on tyres will generally stay in place if the beads are cut, since the bead is trapped in position under the hook of the rim (I'll ignore archaic non-hook bead rims). A severe kink in the bead can prevent it seating and then it will pop off - common when people try to coil wire bead tyres for posting and mess it up, or with bad batches of Schwalbe Marathons.That's a very interesting question.
Most people think tyres are held in place, against air pressure inside, by the strength of the bead. In other words, they imagine that the bead resists expansion as the tyre inflates. Following on this, they then ask questions about the relative strengths of steel and rope (kevlar) beads.
Tyres are not held in place by the bead resisting expansion. The bead is merely there to position the tyre in a groove inside the rim.
The tyre is designed in such a way that it shrinks with inflation and it is thus held onto the rim with a constricting force. You can cut the bead in several places and provided you didn't damage the casing of the tyre itself, you can safely inflate it and ride it. It will not pop off or explode.