Yes, basically O.K. As said, you would want to use a precision OP for the input stage. "Cold junction compensation" is a dummy yet, it should inject a temperature dependent voltage with 40 µV/K slope input related, respectively 10 mV/K after the preamp. Or use any available digital thermometer and perform the compensation in the digital domain.
The circuit of post#6 doesn't include an cold junction compensation.
Thus the output accuracy is wrong by the cold junction temperature.
And the output is as precise as the cold junction temperature.
All in all not a very precise measurement method.
Thus your Opamp doesn't need better specification than the method.
This means: you don' need an Opamp with better than 25°C or about 1mV
And the drift doesn't need to be better than 40 uV/K.
You can safe money by omitting the second Opamp, because it is useless in your circuit.