You can detect radiation in the near field, no problem with that. People do it routinely.
You are assuming, then, that the field of your point source is very nearly zero very far away, at very early times. But this is obviously a non-local condition. Whether it is satisfied cannot be determined by inspecting only the immediate (spacetime) vicinity of your source.
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To this I add, by the way, that Dirac and others have noted that a classical theory of point electrons -- even when augmented by suitable non-local boundary conditions -- is essentially untenable. A consistent classical theory of radiating charges has yet to be formulated.
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Why should I need to assume anything about the field at a distance greater than c t? (You seem more concerned about the mathematical structure of the theory than about the Physics the theory strives to describe. I am not)
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