Why do people use multi stage amplifiers instead of just one
The short answer is that there isn't a single stage amplifier that remotely approaches the ideal voltage amplifier.
An approximation of the ideal voltage amplifier is nearly linear for large signals and has high input impedance, low output impedance, and wide bandwidth.
There are three configurations for single stage amplifiers: common-emitter, common-collector, and common-base.
The common-collector is quite linear, has high input impedance, low input impedance and wide bandwidth. It also has less than unity voltage gain so it simply isn't useful as a voltage amplifier.
The common-base has high voltage gain and high bandwidth but very low input impedance and moderately high output impedance so it's not a good approximation either.
Finally, the common-emitter has high voltage gain, moderate input impedance, moderately high output impedance, and moderate bandwidth. Overall, it's the best choice for voltage amplification.
But, importantly, it's far from ideal. Multi-stage amplifiers can get much closer to approximating the ideal voltage amplifier.
For example, to get low output impedance, a common-collector stage can follow the common-emitter. To get high input impedance, a common-collector can precede the common-emitter.
A more sophisticated approach would be to cascade two common-emitter stages to get enormous voltage gain and then use negative feedback to get the voltage gain down to the desired level. Remarkably, the negative feedback also lowers the output impedance and increases the input impedance all the while improving the linearity for large signals and extending the bandwidth.