A time variant current through a coil wound around a toroid core will produce a time variant magnetic field within the volume of the toroid, but no field external to that volume. Since a time variant magnetic field propagating in space must be accompanied by a time variant electric field to propagate, there can be no resultant time variant electric field external to the toroid volume either.
Now, make a toroidal transformer by threading a loop of wire through the "donut hole" in the toroid and you will find that an EMF is generated in this secondary loop. If there is no electric or magnetic field in the location of the electrons in the secondary loop, what causes them to move? Distances are too great for the Nuclear strong or weak forces to be in effect and gravitation is too weak; so if there is no electric or magnetic field - what else is there to physically explain the production of the EMF in the secondary loop?
And yes, I understand the mathematics of a time variant field contained within the area bound by the secondary loop - but that is not the issue here - is it!
If a physical field is the mechanism by which a particle at one location (electron in the primary wire) changes the behavior of a second particle at another distant location (electron in the secondary wire), and if there is no field, what is causing the effect? How does the electron "know" it should do anything? What accounts for the motion produced? What caries the information to the secondary wire electrons regarding the movement of the primary wire electrons?
Are we to see physical fields as being merely mathematical constructs with no physical reality behind them? Is there no intuitively understandable physical explanation and reality to "action at a distance"?
To clarify the questions being asked:
1) Is an electromagnetic field a real physical entity or just a mathematical construct - a mere map of action potential vs position without any explanation of cause?
2) Is it not paradoxical to state that an EM field is the means by which one charged particle influences another and to then also state that field theory does not require that the field actually exist at the location of the second particle - ie. no local cause. (Example : the electrons in the secondary wire that are being influenced by the electrons in the primary wire of a toroidal coil? EM theory only requires that there be a changing magnetic field somewhere within the area bounded by the secondary loop, not at the actual location of the secondary loop.)
3) Does modern physics and field theory in particular require the abandonment of local causation?