I assume Vcc is supposed to remain constant. Then current through the load should remain constant when the load resistance is changed as long as the power supply voltage is high enough (per Ohm's Law) and as long as enough voltage is available. and not too much voltage is dropped across Re.
Transistors are current amplifiers. In the linear range, Collector-Emitter current is a multiple of base current. But in digital applications we often use transistors (and MOSFETs) as a switch (on/off) and in analog applications it's more common to use a circuit results in a voltage amplifier.