Why can't electronic systems use constant current power source and vary the voltage as power consumption varies?

While there's no reason why you couldn't design electronics with constant current sources, there's a few good reasons why we don't. Batteries and mains power are supplied more as constant voltage sources than constant current sources so it's simply more convenient to use what we have. The other reason is that a constant current source is always burning power. To shut a device off, you would have to short-circuit the power-supply. Since most wires have resistance, you'd constantly be wasting power.

Most power sources are constant voltage, and not constant current. If you take the two main sources of electrical energy, which are batteries and rotating generators(regardless of size), the one thing in common is that their voltage is fixed theoretically to a certain value and can be controlled. For example, a standard AA dry-cell battery has a voltage of 1.5 V, which it will always produce more or less (disregarding real-life errors). The internal chemistry of most batteries relates the internal chemical reactions to the output voltage of the battery. Similarly the generator, for a given magnetic field strength (called excitation), and a given speed, will produce a fixed voltage at its terminals(again, only approximately due to real-life).

In almost any electricity-using device, in almost most cases, a voltage is the cause, and the current is the effect. Only when you apply a voltage to a device, may current start flowing through it (superconductors not-withstanding). Even constant current devices monitor the current and regulate the voltage as per the load. You never hear of a 3 V flashlight battery monitoring the voltage at its terminals. This is due to basic physics, in which change in movement of electrons (i.e. current) is possible when electric field (i.e. voltage) is applied.

The constant voltage method provides the key to load-line design of amplifiers. Given the vast need for amplifiers, the ability to provide stable gain for boosting long-distance phoneline signal strengths, the need for low distortion operation of vacuum tubes in those phoneline circuits, we may feel hobbled by the successes of BellLabs. But then Bell provided the transistor. Here is a load-line.