I answered your other post, but there is new information here, but it doesn't change my answer. If the oil system is clogged up by sludge, the problems occur at high rpm, not low. An oil light at idle means a bad oil pump or bad bearings. This engine has a history with the latter.
The water pump is driven by the timing chain, so the theory is that if the water pump were to "freeze up", it would break the chain. This is highly unlikely if the car is driven regularly. On cars with timing belts that drive the water pump, a leak could cause the belt to slip so the water pump replacement is a little more valid, they are more likely to leak than to freeze up.
The timing chain itself is generally reliable, the tensioner is the weak point in this engine, but that is not the cause of the oil light. To get to the oil pump, the timing chain is one of the components that has to come out, so you might as well replace it, and the water pump while your at it and definitely the tensioner. With that much work, you might as well go for a complete reman, except this engine doesn't rebuild easily so a good reman is hard to get and very expensive. Do consider that 3.5 instead.
I still recommend switching to 15w40 diesel oil and run the engine till it quits. I had a neighbor get over 200k on his 2.7, the timing chain tensioner got him in the end. His was a gen I engine, he rarely changed his oil, but being a diesel mechanic, guess which oil he did use.