why is there no electric field at the center of a charged spherical conductor
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The thing is that has the logic not quite correct. (A) The electric field is not zero because (B) all of the charge is on the exterior. Both (A) and (B) are true because of the property of conductors.
If there existed a potential difference across the interior of a conductor, then electrons would move to neutralize that potential difference, and indeed they could do so because it is a conductor. And if there were a potential difference, then there must be a non-zero electric field.
But precisely because with a conductor electrons can move and eliminate all internal potential differences, there is no electric field and by the same logic together with Gauss' Law, all of the charge is on the exterior of the conductor.
If you'd like to see this worked out in more rigor, watch this:
@JamesJ: "because of the property of conductors" doesn't quite sink well...the only property of conductors is that electrons/charge can move freely in it..its every other property, related to electricity, is more or less a consequence of that only..
electrons can move and eliminate all internal potential differences that is exactly what i meant too..