Pinch off in MOSFETs
doesn't it mean that the source and drain are now isolated (since the channel has moved away from drain) and Id becomes zero not constant?

You seem to assume that the depletion layer prevents current to flow but it doesn't. What then happens is that the size of the channel is determined by the gate-source voltage (assuming source is connected to bulk). This also determines the maximum drain current that can flow. Any current (electrons) "escaping" the channel at the drain side leave the channel but enter the drain's depletion zone and are drawn to the drain as it will have a high positive voltage.

The drain can only prevent current from flowing by having a low voltage, zero actually. Note that in that case it will also not have a depletion layer anymore, since it has no positive voltage.

For a depletion layer to form the drain must have some positive voltage so current will flow in that case. For Ids to become zero either Vgs must become zero so that the channel disappears or Vdrain must be zero like I just explained.

The total depletion layer has a neutral charge but only when no current flows. In the discussion above the channel injects electrons into that depletion layer so there will be mobile carriers present. I suggest that you go back to the chapter about the PN junction in your textbook and browse through that again. It is complex, it takes a while to understand this. I know it took me a while.