It sounds to me like something on the EFI circuit is pulling too much current. Perhaps a wire somewhere is pinched or maybe the fuel pump windings have a short??? Regardless of what it is, it sounds like it's caused some damage on the supply side of the EFI fuse. Instead of finding and fixing the root problem, it sounds like somebody made a jumper wire to power up the EFI circuit off the oil level relay (not the best plan). Evidently the short (whatever it is) isn't strong enough to prevent it from functioning, just makes it pull enough power to create more problems. I believe the stalling occurs when the jumper gets too hot, then goes away when it cools down a bit (or arcs just right and holds for a bit longer).

For this type of problem I would use the jumper wire just long enough to track the high amp draw. Use a DC amp clamp to determine how many amps are there. If it's more than 5 then check for amp draw under the van where the power wire goes to the fuel pump. Check as close to the tank as you can get. If you're seeing excessive amps here then it's a bad fuel pump. If not then you'll need to track the harness backwards until you find the place that's eating all the amps. After that's repaired you'll need to find the damage to the EFI fuse supply circuit.

I despise "hack repairs" and "work-arounds" As soon as you determine the cause I would find a Previa in the salvage yard and get the parts you need to put it back to stock. Good luck. Tim