The spring can matter. The pictured hydraulic valve has a screen on the diaphragm assembly, which is indicative of a normally-closed valve. That would mean the pressure above the diaphragm would equal that below the diaphragm, so it isn't water pressure that would close the valve. It would be the fact that the upper-chamber diaphragm area is greater than the area of the diaphragm assembly where the seat washer is. The hydraulic force is a function of the surface area the pressure is acting against. The difference in surface area translates to a force that holds the valve closed.
When a diaphragm valve gets stuck wide open, the pressure is just about equal on both sides of the diaphragm, over areas that are just about equal on both sides of the diaphragm, and all the while the rushing water itself presents some additional force in the way of its momentum. A flow control can make all the difference here, because introducing a slight pressure drop creates less hydraulic force on the inlet side of the valve diaphragm, and the valve can close. Absent a flow control, and absent any spring force inherent in a flexible diaphragm itself, it becomes the job of the valve spring to provide a force that allows the valve to close.