the chain tensioner is pumped by oil pressure but also holds by a ratchet pawl once the pressure is removed - That ratchet pawl can wear out and cause very rough startup noise and even catastrophic failure if the chain slacks and causes a slip on the cam timing (not quite common with roller chains like mercedes, usually seen with morse chains)

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2012 BMW X5 (Beef + Granite suspension model)

1995 E300D - The original humming machine (consumed by Flood 2017)
2000 E320 - The evolution (consumed by flood 2017)

how your engine hanging? How many miles since then?
Mine was making the same sound which disappeared when rev was slightly increased from the idle speed when running on 5w30 oil.
After the 5w40 oil change i am getting no audible noise, only using a stethoscope i can hear the same noise.

The sympthoms are the same as yours.