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Then find an audio amplifier IC. A low power amplifier IC might have a couple decoupling capactors and a resistor or two. A single transistor amplifier could have a dozen or more supporting components.
It is possible to use a N channel power mosfet, connect the 4 ohm or 8 ohm speaker right between the drain and the + power supply lead. Biasing the gate to class A operation you could get some power out that way and it would be pretty sensitive.
I simulated your small amplifier.
1) With an 8 ohm speaker, it doesn't have gain, it has loss.
2) It has a very small output power, a whopping 0.8mW. Headphones have much more power.
3) It is distorted.
I also tried it as an emitter-follower and its loss is less. Its output power is 0.2mW.
I used a 2N3904 transistor in the sim with a typical current gain of 230.
I also tried a 2N5089 that has a typical current gain of nearly double and the Q-point shifted towards saturation but the voltage gain increased only a little:
The open loop gain of the transistor is actually Rload/Re. Re is .026/Ie. Ie, for a beta of 100, is about 8.72mA. This makes Re=3, and Avol=157. However...
You forgot about Rin which is approx. Re*Hfe, or about 300 ohms. This is significant. I'll let you do the math, but below is the transistor amp and the equivalent circuit. The gain of both is 2.3 in simulation. With a 2N3904 in the same circuit, I also got a gain of 3.35. The higher beta lowers the collector voltage, increasing Ie and reducing Re. This increases Avol, which helps a little in this case, but Rin increases by a factor of 2 or more, which helps a lot.