Grumpy_Mike:
but you'd have to make sure the BJT is biased so it is controlled by its base-emitter voltage, and not the base-emitter current
So do we ensure this by rewriting physics?
Two words: "Voltage buffers". In that type of circuit the current (while still necessary to operate the BJT) can be set arbitrarily to a fairly small level and VIN determines VOUT. Unless you think a NPN common collector circuit requires non-standard physics... 
Edit: Perhaps I overstated things a little by using the words "have to" instead of "should, and Grumpy_Mike is certainly correct that 3 mA can control a current hundreds of times larger. However, it is entirely possible to use a BJT so that one voltage controls another without violating the known laws of physics!