In real world, signals are always finite, which means that any aperiodic signal can be periodic signal by repeating themselves. Then, why we don't just use Fourier series for those finite aperiodic signals?
You have a good point about any real signal (and any real piece of signal processing equipment, for that matter) having a finite lifetime.
However, if you don't know what that lifetime is ahead of time using the Fourier series becomes complicated. Basically, rather than mess around with the artifacts caused by the frequency domain being sampled in time, you just approximate the signals and systems involved as existing for an infinite amount of time.
As long as the amount of time the signals stay on far exceeds any time constants in the signal processing circuitry or algorithms, then the approximation is a good one, and works more than well enough.
I believe that the premise of your question is wrong. In practice we almost exclusively compute the Fourier series of finite length signals. In signal processing we mostly deal with discrete-time signals, and we use the Discrete Fourier Transform (DFT) - usually implemented by the Fast Fourier Transform (FFT) - for transforming finite portions of these signals to the frequency domain. The DFT coefficients are just the coefficients of the discrete Fourier series of these finite length signals.