Why is transistor gain measured in terms of current?
The inherent ratio of emitter/base current is BETA. This ratio does not describe the voltage-gain ratio. BETA is important for biasing, and computing input resistance (the amount of energy demanded from the signal source), but does not set the voltage gain.

Many electronic active elements have a transconductance, abbreviated gm.
Multiplying [gm * output resistance] does describe the voltage-gain. This math is accurate for bipolars, FETs (MOS and J FETs), and vacuum tubes.

For bipolar devices, gm is Ie/0.026; at 26mA (0.026 amps), gm is 0.026/0.026 = 1 amp/volt. At 26 microAmps, the gm is 0.000026/0.026 = 0.001.

A bipolar biased at 26uA, with 10Kohm in the collector, will have 0.001 * 10K or voltage-gain of 10x (20dB).

To achieve controlled voltage gain, external resistors (one in collector, at least; optionally one in emitter without bypass capacitor) are used.

Maximum voltage gain for this CommonEmitter circuit is VDD / 0.026volts. Thus 9v supply / 0.026 could produce 9*39 = 350X, with challenging biasing. If collector resistor were replaced with constant current source (an PNP), your gain can rise another 10X or 100X.







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Here are two bipolar stages, biased identically. Left has Av = 1.0X, right has Av = 18X.







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