The problem is one of semantics.
"Pinched off" does not mean pinched closed. Instead it's a technical term; the label for an operating mode. "Pinchoff" means "device operates in constant-current mode." We could have chosen a different term, such as BJT's saturation/cut-off, but now we're stuck with it.
First, with the gate voltage reduced and the channel wide open, the channel has minimum resistance, with near zero voltage between drain and source; acting like a short circuit. Next, increase the gate voltage, so the depletion region invades the channel from the side. The channel behaves like a physically narrow material. It's resistance increases.
With a drain resistor present and gate-voltage increasing, voltage along the channel increases with increasing channel resistance. Finally it grows large enough to rival the gate voltage which produced the depletion region. At that point the linear "resistor behavior" becomes nonlinear, and the channel begins to behave like a constant current source, rather than a simple resistor. The constant-current mode is called "Pinch-off." But the channel is only narrow, not pinched closed or "off."
To remove confusion, perhaps personally label it "nonlinear mode?" "FlatPartOfTheGraph?" Other better labels?
During constant-current mode, the narrowed channel has gone into breakdown, and it's being shaped by negative feedback processes taking place between different regions along the channel. (If it was positive feedback, the channel would exhibit instability, "turbulence" and become an oscillator.)
With the channel narrowed and breaking down, if Vds is then increased, the pattern of channel-voltage and the borders of the depletion zone dynamically change, causing the narrow channel to become longer. Longer channel has higher resistance, with a self-regulating effect that produces a constant current independent of Vds changes.
Heh, increase the gate voltage sufficiently, the channel narrows to nothing, and the drain current falls to zero. The device enters "pinch-closed" mode!
Is there a hydraulic analogy? How about two balloons being pushed together. If we blow compressed air between a pair of balloons, that's a pair of depletion zones with a conductive channel between. Push the balloons together and, when they seem to be nearly touching, they'll distort, and make a loud raspberry sound. That's an unstable channel. A real FET channel wouldn't oscillate, instead it would remain narrow, with flattened balloons forming its parallel walls. Increase the compressed air feed, and the balloons change shape, making the channel longer, with length increasing just enough to keep the flow constant. Now push the balloons together harder (higher Vgs,) and the channel becomes uniformly narrower. Pushing the balloons 'programs' the constant-current effect for lower current; decreasing the flow.
For balloons, a higher "Vds" pressure-feed can never force a larger "Id" flow through the channel. Instead the balloons distort, widening the flat region, which increases the channel length proportional to increased Vds pressure. At some voltage the balloons would rupture. A Vds punch-through.
I wonder if real balloons would give a constant-Id characteristic? Even if they were making raspberry-sound oscillations, increasing the compressed air pressure would change their shape and their oscillation, and might end up lengthening the channel, preventing increased flow. Go and measure the loud, wobbulating balloons' Vds pressure, Id flow, and Vgs pressure. Ignore the unstable AC part, and you might be able to sweep the pressures and experimentally obtain a set of characteristic curves for the very first Field Effect Balloonistor (FEB.)