"The magnetic field around the wire doesnt just suddenly appear, it progressively grows stronger as the magnet comes closer"
Initially the magnet is at rest. Now there is a certain amount of magnetic flux linking the coil.
But since the flux is constant there is no current induced.
Now suppose that we start to move the magnet at instant t=t1. The flux linking the coil will increase since the magnet is moving closer. The question is when does it start to increase?
Certainly it cannot be t1 since that would be mean faster than light information travel.
So there should be a delay and will it not be equal to the time taken for the effect of the changing magnetic field to reach the coil.
As an example, if the sun were to disappear at this instant of time, we would not notice it for the next eight minutes after which we will plunge into darkness.