I have a Shimano Nexus 3 speed with a coaster (and hand brakes) that is my daily driver, and where the coaster brake catches is pretty random. I can sometimes make several, as in 3 - 5, full backwards pedal rotations before it catches, other times it catches immediately. I am now in the habit of catching the brake before I descend any large hill. Since I live in Nobb Hill SF, that's every day.
Of course Schwinn today is not the same as they were then.
I have never had a coaster brake failure. Nor have I experienced a problem with engaging the coaster brake as you describe. I have always found the single coaster brake to be more than adequate and to be more reliable than any (set of) handbrake(s). I like the fact that I can skid the rear tire if I apply enough force. I live in a non-hilly area, so YMMV.
What's unusual is having a disc brake as the second brake, but I like those much more than rim brakes.