There are several Sturmey Archer coaster brake hubs. While they may function normally as 3 speed hubs, at least a couple of the coaster brake modeal are known to have serious braking defects. This seems true with the one I own (a TCW), and on the others I've tried. There are a couple problems with the brakes on these. One is that they just are pretty crappy as far as coaster brakes go. A bigger poroblem is that when braking, particularly in high gear, is even less effective because the brake is also coupled through high gear. This is kinda like giving a already poor brake even worse mechanical advantage. And a even nore dangerous problem is that these hubs, like the more common freewheel 3 speeds, can go into "neutral" between 2nd and 3rd if the cable is not adjusted correctly, which can be bad enough, but also disengages the braking ability completely, since the braking is linked through the gear train.
These hubs can still be useful if you also have at least a front brake on the bike. My Sturmey Archer coaster brake bike has a caliper on the front which helps stopping capability of the bike 300%. A rear caliper is probably not a bad idea, either. Would be silly if the coaster brake was more effective. But in this instance, it makes more sense.
Aside from the brakes, the one I own works fine. In fact, I've considered trying to disable the coaster brake completely and effectively turn it into a freewheel 3 speed. However, I've never taken one apart to observe how difficult this would be.
There does exist a AWC coaster brake model. This is a more recent design. I'm not sure when it came into being, but it's supposed to be much better than the older designs, addressing all the old problems, basically being a coaster brake version of the old reliable freewheel 3 speed, the AW. However, I've never tried or even seen these models.