That sure sounds like a thermostat problem if everything is cooling off. The radiator thing is very suspicious though, the replacement radiator has one less coolant line that tees off of the hose from around the water pump/thermostat area and goes to the oil cooler. The change up hose to go with the updated radiator has the branch blocked off. Maybe something got messed up there??
I don't think there is a auxiliary water pump on the E70 but I haven't really checked into that.
OK I didn't read that right, you had heat the winter after the rad was replaced so that makes it less likely to be the problem.

You didn't mention anything about codes??

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1988 325is (purchased new) sold
2004 X5 3.0 2005 X3 2.5
2008 X5 3.0 (new to me)