Confusion of how current can flow in a 1+ henry LR circuit
No there is no instant current.

I starts at 0 with dI/dt=V/L when you connect a battery across a large coil regardless of  Rs.

I put an 18650 Li Ion battery directly on a 22 Henry inductor ( the primary of a 5MVA transformer) intended for 30kV.  The result was measured with a DMM DC current mode.  

It was, as expected, with 3.8V across it starting from 0A and rising slowly at dI/dt=V/L= 138 mA per second.  After a minute , I disconnected and got a nice long arc around 8 Amps that lasted for a while.  The ESR of the arc depends on the current and the current density at contact, but I had stored a lot energy.