Wanted to update everyone on this.
Took my car, 2014 GTI with 92,000 mostly highway miles, back to the dealer and they were able to duplicate the temperature rise and the diagnosis was a bad water pump. Mind you, my water pump was already replaced under recall around 55k for leakage. This time around there was no leakage, nothing to indicate a cooling issue other than the temp rise.
Told the dealer to go ahead and replace it and they gave me a loaner. In the meantime I talked to a local European auto repair place and they said that they've seen the TSI water pump impellers start slipping from the shaft. Not sure if the impellers are keyed into the shaft with a keyway, or how the two components are held together. But the implication was that the shaft may be turning but the impeller wasn't, or was turning slower than it should due to the slippage.
Anyway, $1180 later I got my car back. I asked the dealer to break down the cost for me and it went roughly as follows: 5 hours labor x $140/hr = $700. Water pump, I think ~$200. Coolant flush and refill, $150. Then some miscellaneous parts and tax brought it up to the miserable dollar amount shown above.
You may be interested in knowing that my P2568 code has disappeared with the replacement of the water pump. I like my tech at the dealership, he warned me long ago to change the oil every 5K because of the timing tensioner failures. His initial diagnosis of the P2568 (after calling it a mystery because nobody had ever seen it) was the time it takes the coolant to warm up was not within spec. He said there is a calculation in the ECU that has to do with that warm up period and if it isn't met, it trips the light.
The problem is determining why that is happening. If my water pump was acting up, as in slipping like the other place I called said, maybe low coolant flow was leading to improper warm-up times?
Most of you know more than I do about the ins and outs of the VW ECU, and maybe there is a coolant flow sensor, though if I had to guess that may only be to detect if there is coolant flow (yes/no) and not actually measuring the flow rate being delivered by the pump. So the ECU may have been detecting some flow, but the pump wasn't delivering enough.