Using OP-AMP buffers as shown in ther original post is a bad idea, because they add large offsets to the thermocouple output voltage.
I don't see how CMOS analog multiplexers would add large temperature errors. MAX31850 has specified input currents < 100 nA, so multiplexers with < 100 ohm on-resistance achieve < 10 µV voltage drops.
The cold junction compensation is inside the MAX31850, the design must assure that the input terminals have the same temperature as the chip.