Let's start with a little digression: 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.
You should also ask yourself why you need a base resistor. Try to control the BJT by removing that resistor and by supplying a pure voltage (that is voltage from an ideal generator) between base and emitter. Not so easy, uh? (Conversely, with a MOSFET you'll have trouble in trying to use a current generator...)
The bottom line is that it's easier to distinguish between 10 e 40 uA than it is to separate 0.65386 and 0.65389 V.
Note: The dependence on Vbe shown in the Ebers-Moll model is not implying a cause-effect relationship. It's just simpler to write the equations in that way. Nobody forbids you from using inverse functions.