An irrigation valve normally has five major parts: an inlet and outlet chambers, a diaphragm chamber (d.chmb), a solenoid and a spring which provides additional closing pressure onto the diaphragm. There are also two small vents (openings); one (input) which connects the inlet to the d.chmb and a second one (output) which connects d.chmb to the outlet through the solenoid (whereas solenoid can block the opening while off and holds it open while on).
This configuration normally allows two configurations: While valve is off, the output vent is blocked and input vent allows d.chmb to equalize with inlet chamber. This, in turn, allows the spring actually exert enough force to close the valve. While the valve is on, the output vent is unblocked allowing the water to flow through the d.chmb freely thereby reducing pressure relative to the inlet chamber. As soon as the pressure difference between inlet chamber and d.chmb is greater than the force exerted by the spring, valve opens up and continues to operate until output vent is blocked and d.chmb pressure is equalized with the input.
I believe that the reason for 5 second issue you’ve experienced is that you are using a well pump, meaning that there is a period of time whereas irrigation pressure is constantly increasing while the pump is ramping up. Due to the size limitation of the input vent the pressure within d.chmb will be lower than the input chamber until sufficient time to equalize has passed.
As far as I know, the flow control on your valve is connected to the spring that provides a closing force. The more flow control turned in a CW direction, the stronger that spring becomes (the knob turns the screw which, in turn, pushes the spring harder against the diaphragm). Pushing the spring tighter against the diaphragm allows you to essentially overcome the pressure differences during the pump startup.
It is possible that the flow control screw on that particular valve became loose with time, or something got stuck in the input vent thereby increasing the time d.chmb would take to equalize with the input. It is also possible that it is simply a question of proximity of the valve to the well pump.
Here is a better video on how the valve works, in case I confused everyone 