It's not the "class" of sensor that makes the difference, it's the specific sensor itself (the multitude of Nikon and Canon DSLR models with significant variance in quality should illustrate this). Also, APS-C isn't a specific size, it's a classification that different manufacturers interpret differently.

More importantly, though, I would say that "plenty" is a stretch. There might be some with large enough sensors, but those models are still the exception, not the rule, and most likely they will be at the top end of the price point for a compact anyway... I mean, the Coolpix A is $1000+. You can get an entry level DSLR or a pretty good used one for about that price. You're actually paying more for convenience than quality, at that point.