3800 series 3 coolant

Your research is correct. Coolant does not flow through the timing chain cover. Oil is present but not coolant. Since the water pump is in the same general area, your suspicion may be correct and you need another water pump. Take the car to a different shop and don't tell them what the first shop said (until he has provided a diagnosis and quote to fix the real problem). Good luck.

Have you checked the plastic elbows that take coolant to the heater core? They often fail and will dump all your coolant before you know it. It is highly unlikely that you need another water pump if you replaced yours a year ago. A burst hose is also a possibility although that's pretty rare these days too.