Trying to understand the relationship between voltage, current, and power
I'm assuming we are talking in terms of DC here to keep things simple. 

Voltage is a measure of how much a single electron's energy changes across the circuit.  Current is how many electrons are flowing through a circuit.  Multiply those two values to get the rate of energy transfer, or power.  If all you have is a measure of current, you know how many electrons are flowing through the circuit, but not how much their energy is changing.

You are correct that voltage induces current, and that voltage and current are interrelated.  Ohm's law is the simplest case of that.  You can derive a circuit's power from current and it's resistance using known formulas.  And even more simply, if you know current is flowing you know the circuit is dissipating power, even if you don't know how much (as the other commenters have said)