Observations of binary pulsars and pulsars in globular clusters suggest that at least some pulsars must receive weak natal kicks at birth. If all pulsars received strong natal kicks above 50 km s(-1), those born in globular clusters would predominantly escape, while wide binaries would be disrupted. On the other hand, observations of transverse velocities of isolated radio pulsars indicate that only 5 +/- 2% have velocities below 50 km s(-1). We explore this apparent tension with rapid binary population synthesis modeling. We propose a model in which supernovae with characteristically low natal kicks (e.g., electron-capture supernovae) only occur if the progenitor star has been stripped via binary interaction with a companion. We show that this model naturally reproduces the observed pulsar speed distribution and without reducing the predicted merging double neutron star yield. We estimate that the zero-age main-sequence mass range for noninteracting progenitors of electron-capture supernovae should be no wider than approximate to 0.2 M (circle dot).