Correct , the performance is not good because the valve "saturates", i.e. reaches a physical limitation of fully closed, at times during the transient.  When the valve is completely closed, feedback control is no longer possible.

Windup does not appear to take place, because as soon as the controlled variable approaches its set point and the valve should open, it does open.  If windup occurred, a substantial delay would occur between the time that the error changed sign and the controller output (and valve) moved.