Yes, work hardening increases the tensile strength. No, it doesn't have to be done repeatedly to cause brittle failure. When you work harden steel, it basically horizontally shortens and vertically stretches the material's stress-strain curve. It takes less deflection to fracture, but the applied stress to cause a deformation or fracture increases. Realistically, you don't apply too much stress to the handlebars unless you are doing artistic cycling or riding your bike over open sewer holes while doing artistic cycling, so it isn't of much concern.