I would start with checking for trouble codes. Next I would clean the throttle body and the IAC. If carbon builds up it can cause idle issues and the IAC(idle air control) can cause poor idle /no start. These also can have misfire issues caused from the coil failing . Very common for my dealership to replace the intake manifold gaskets, spark plug wires and the coil pack. we get many windstars
I would say check the fuel pressure and see if there is a bleed down issue but after 4 pumps I hope they got it right and fuel is not an issue. It would be nice to do an injector flow test to make sure no injector bleeding down causing the long start.
A scanner with live data really makes the difference with diagnosis
let me know what has been replaced or services have been done