What makes a generator a current generator instead of a voltage generator? Look at the V-I characteristics: the one with mostly constant voltage (almost horizontal in the I-V plane) will be called a voltage generator, the one with mostly constant current (almost horizontal in the V-I plane) will be called current generator.
This is because the 'accent' is on the constant quantity (the voltage or current supplied - the other variable depends on the load and the compliance of the generator).
In a controlled device, the accent is on the variable quantity. Given the exponential input characteristic, that leaves Vbe almost constant, it is current you like to see as the controlling variable.
The bottom line is that it's easier to distinguish between 10 e 40 uA than it is to separate 0.6538 and 0.6542 V.