Why not change magnetic flux only at the edges?

I learnt Faraday's Law of Induction however I was confused about why the area aspect of magnetic flux impacts on the EMF generated.

Why do magnetic field lines inside the coil area affect the EMF generation? I was taught that the reasoning behind Faraday's Law was that the relative motion between the magnetic field lines and the coil invoked the Lorentz force on the electrons, pushing them into a current. So why should the magnetic flux through the center of the coil do anything? Only the magnetic flux change AT the wire itself should be inducing current right?

So I thought of the experiment below. In the diagram below (sorry it's so poor quality), the black circle is a solid loop of copper wire and the red X's are solenoids. If I turn on the solenoids, shouldn't I get an EMF of equal magnitude compared as if I put the coil into a bigger but equal magnitude magnetic field? • If I turn on the solenoids, shouldn't I get an EMF of equal magnitude compared as if I put the coil into a bigger but equal magnitude magnetic field? It's not clear how your solenoids are set. If the total flux within the coil is less than $BS$ (as I suppose) the emf will also be less. – Elio Fabri Nov 24 '18 at 16:15
• Think instead of putting just one long and thin solenoid (section $S'$) at the coil's center. If it produces a field $B'$ such that $B'S'=BS$ the emf will be same, although the field is zero at the coil. This is because the thin solenoid's flux, when varying, will generate an electric field all around it, of intensity decreasing as $1/r$ so that the induced emf would be the same in any loop circling the solenoid. – Elio Fabri Nov 24 '18 at 16:17