Voltage of thermocouple
The current (if terminated in some kind of load or a short) will be dependent on the resistance of the thermocouple wire and connections, which will vary with the temperature along the length of the wires, the soundness of the connections, diameter of the wires and so on. 

The voltage is only dependent on the temperatures at the hot and cold junctions and the wire composition. 

Of course if you measure the current into a resistance that is much larger than the thermocouple resistance, then you are effectively measuring the voltage and it will work well. 

You can still find self-powered temperature gauges that use a meter movement that operates directly from the thermocouple, and thus measures the current through a copper coil and the thermocouple wire. To achieve accuracy they're sometimes marked with the nominal resistance of the thermocouple that they expect (eg. 10\$\Omega\$). They need temperature compensation for the positive TCR of copper and some kind of bimetal mechanism for cold-junction compensation.