The simple four-resistor-plus-op-amp differential amplifier does have a CMRR limited by the resistor matching.
There is going to be confusion if confusing terms are used. We should at least agree on definitions.
Differential amplifiers have differential inputs and outputs. Difference and instrumentation amplifiers have differential inputs and single ended outputs. Instrumentation amplifiers have high impedance inputs which minimize the effects of source impedance.
Texas Instruments among others likes to confuse the issue, but resist it.
I don't find it confusing, though I usually use the term 'fully differential amplifier' for the amplifier that has differential input and output (even though usually those are high speed).
So why is CMR an issue? Let's say you put together an amplifier out of 1% resistors, and it has a CMR of -40dB. You want to use this amplifier to amplify 1mV signal by 1000. If your input signals are 0V and 1mV, your output is 1V, everything nice. Let's say your input is 1V and 1.001V. Your expected output is 1V, but because of resistor mismatch, you get 11V (-40dB is 1/100). And this might happen at DC, or it picks up 50Hz noise, or your signal has other sources of common mode noise. There are ways to improve this, for example ferrite cores and placing your wires through it, but ultimately, it's the best to use very much matched resistors.
And you can either select these by hand, get expensive 0.001% resistors, or get one of these nice ICs, where these are internally matched, often times laser or electronically trimmed. But ultimately it's characterized, which is important. You don't get surprised, when the output is 11V, because the IC has -100dB CMR.
And you might be familiar with this, and maybe there is no new information for you in my post.
I have been considering to use AD8129 with their evaluation board:
https://www.analog.com/media/en/technical-documentation/user-guides/UG-133.pdf
AD8129 is not probably lowest noise, but it might do the job?
Looks like a very useful board for this purpose.