It's more likely the pump kicking off dues to temperature. I've seen it a few times.
My juniors Honda Accord was one. The pump would run ( although a bit noisey ) for an hour or so. The engine didn't run great but drivable until ithe pump kicked off.
After a few minutes the car would start and run like crap for maybe 30 seconds. Let it sit overnight and go driving again.
New pump cured it.