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The coolant hoses bring the coolant to the oil cooler to take the heat out of the oil. So it wouldn’t be an actual cooler then just a passage for the oil
IIRC, the reason for the cooler delete was an internal redesign rendered it redundant. So, I don't think it is a good idea to remove it. I would put new oil seals in while it is out of the car along with a new timing set, the oil spigot needs updating at least.
I do wonder why yours is missing? I thought that all 2006 2.4L and some 2007 2.4L had them. Are you sure you have a 2.4L not a 2.2L? You could tell by the VVT embossed on the valve cover. OR by the 8th digit in the VIN; F=2.2L, B=2.4L.
Looking at the pictures of the engine in the back of the pickup, the oil cooler oring are leaking and need to be changed.
That being said, as Donbrew points out check to ensure you have a 2.2 as this 2.4 won’t run if you put it into your HHR.
I replaced the O-Rings in the oil cooler and I had to put a new exhaust manifold on it.(The one on the used engine and the one that came out of my HHR were BOTH cracked.)