Suppose you are on a distant planet "watching" earth with a radio telescope. You just happened to be tuned to the frequency that the International Space Station transmits a spot beam down to Houston on. You don't see it when it is pointed away from you towards earth, nor do you see it when the earth blocks the signal. In fact, you see it only for a moment as it transits around the limb of earth just before it is blocked by the earth itself. You also see it as it reappears around the other limb before it points away again. Since the ISS orbits about once every 90 minutes, you see the signal twice in each orbit or about once every 45 minutes. You of course, know nothing about the ISS nor the fact that it is in orbit so you assume the observed signal is attached to the surface of the object being observed -- earth in this case. Would you conclude that the earth rotates once every 45 minutes or 32 times a day based on your observations? If so, you would be wrong.
What data fixes the observed thermonuclear "flicker" to the pulsar's surface? Are you sure the gasses being compressed into a thermonuclear explosion aren't "in orbit" around the pulsar as they spiral down towards the pulsar's surface?
Well, you probably wouldn't, because you'd see that every other pulse was doppler-shifted in an opposite direction.
This effect would be far more pronounced in the case of a neutron star, where the velocities are hugely greater, and you may also see the orbits precess. The most telling thing, however, is that radio objects in orbit around a neutron star will have different orbits, and therefore different periods, whereas the observed periodicity of pulsar pulses is stable to one part in 10^8!
I might also point out that these rotational periods are at the limit of what the strong nuclear force can hang onto; the period of an object in free-fall at a larger radius is bound to be much smaller.