The paper is careless in its exposition. The idea that Maxwell's equations and conservation of energy require instantaneous propagation of force is most probably wrong.
The equations in (1) are not standard wave equations with charge and current source terms, but instead some part of the field is used as a source. Then the source is distributed in the whole space and this precludes simple interpretation of ##E_w## as EM wave propagating with speed ##c## from charged body.
The examples with the charge and current loop do not support author's claim in any way. The field is often treated as instantaneous function of charge and current distribution not because the field propagates instantaneously, but because the difference from the correct retarded field is negligible for slowly oscillating currents.