BigClive,
I don't think there's anything technically incorrect with what you are saying.
Here is why I wouldn't state things that way - first if I make an engine with high CR relative to low CR, and run the thing without fueling it, the exhaust temperature of the high CR is higher than the low CR. Entropy dictates that less useful work is performed, unless that heat is what you were after.
Second, the efficiency gains are due to the Carnot temperature differences. Remember that, although doubling compression only doubles temperature (even less, actually), it also increases the Carnot efficiency meaning you can get more useful work out of the combustion. There are also some ancillary temperature effects, like reaction rate positive feedback, that drive your peak temperatures higher than might be indicated by the CR alone. So you affect the high side of Carnot with the high CR. You can affect the low side of Carnot with a high ER, but note that for a nominal engine, as stated above, the diesel will result in a higher post ER temperature (all other things being equal such as the same charge temperature and moles). Atkinson tries to go deeper, but that's not what I see as the thrust of this discussion.
OK, so the useful work is available due to Carnot, but you have to recapture it mechanically in some fashion, and that happens during the expansion period, so that could be a way to state this. I just think it's odd to think of this as an expansion ratio effect when the ER relative to CR is the same for a nominal gasoline or diesel. However, as I stated, I'm not sure anything you are saying is technically incorrect.