That's not how it works. I think you are confusing something. For a 40db CMRR, you must have gain matching of gain matching between the two channels to +/- 0.5% (1% total), and you must have perfect transistor matching (which they never are).Look at the picture I provided. Each of two amps operates in non-inverting mode and is referenced to inverting "-" input of another amp. Since voltage at inverting input is the same as voltage at non-inverting input then voltage across R2 is V2-V1. For common mode signal V2=V1 and voltage across R2 is zero. It means you could remove R2 (no current flow) and without it you just have two amplifiers with perfect gain of one each (followers) even if R1s are a little different. R2 plays role only for normal mode signals. That way you can set any gain for this stage and gain for common mode will always be one. If you set gain of 100 then CMRR will be 40dB.
As for the balanced connection - it is not balanced anymore when you connect it to unbalanced input. Noise currents induced in both wires will be different and noise will go thru. The fact that cartridge is floating won't help, unless input is balanced.
The differential input of an op-amp does not have a gain of 1Nobody said that. Instrumentation amp is not an op-amp.