I'm with FarHorizon--sorry, I didn't read the whole post the first time and got fixated on the gearing issue. Coaster brakes are sort of interesting to take apart, but not nearly so much fun to put back together....
You probably could find a better bike in a thrift shop for less than $50, perhaps MUCH less. I recently bought my wife a Specialized HardRock (a medium cheap mountain bike from the early '90s) to ride in the rain, in very good shape, for $20. If you find something like that and don't want to fool with shifting, just put it in a gear you like and leave it there.