2007 Waverunner FX HO overheats when idling

I am struggling with an FXHO that intermittently sounds overheat alarms and falls into limp mode when almost-idling for 5-10 minutes after a normal speed ride of 15+ minutes.

When the alarm went off recently I observed exhaust gas (instead of water) being released from the port side pilot hole while idling in the sea. I disconnected the coolant feed hose from the outer exhaust joint. At idle RPM there was no water flowing out of the feed hose. When i revved up a little, there was a reasonable amount of water coming from the hose.

I took the ski out and fed tap water through the flushing port. There I got plenty of water from the port side pilot and the engine would cool just fine. So it seems like cooling water pressure from the jet pump is too low to push water up to the exhaust pipe jacket, and subsequently to the port side pilot at low RPM. Therefore, exhaust gas is creeping up the exhaust pipe's cooling jacket.

What could be a likely cause for this overheat condition?