If you are serious about running on the beach, yes it is very runnable. Most of the sand is as you said, soft and difficult to run on. But if you run just above the surf line, where the sand is damp and compacted by the water, it is fine for running. This is not difficult to do at all, but you need to watch where you are stepping a bit or you will get wet. If fact, if you run the beach you will most likely get wet feet and shoes. I suggest not wearing new shoes for this reason, and where some low cut sox to protect from getting blisters from the wet shoes.
I love running the beach at CSA in the morning. I try to get out just after sun up, while it is still relatively cool. Run to the south, walk out to the beach and turn left, and you can run about 3 miles one-way down the beach till it gets to the river, or near there anyway. You will see many of the Jamaican folks starting out their day along the beach. You will detect the scents various kinds of breakfasts cooking, the sea air and the distinctive aroma of starting the day Rastafarian style. It is a wonderful slice of the Jamaican culture that far to many tourists never experience. It's all good.