No, they don't. It's not just the CPU, it's also that phones will often take bracketed exposures and use sensor data to align them correctly and automatically create an HDR image, for example. Either that or taking lots of short exposures and stacking them. It's not just in post processing, the magic already happens while taking the image itself.

People who think digital cameras do anything close to what modern phones do don't have a clue. They're the opposite of 'Phones can do anything a DSLR can' people and they are just as wrong.