As we know, both the DTFS (discrete-time Fourier series) and the DFT (discrete Fourier transform) are used to represent discrete-time periodic signals for all time (or the periodic extension of aperiodic signals in the time interval where the DTFS or DFT is computed) using a finite number of different harmonically related real discrete sinusoids or complex exponentials. The only difference between the DTFS and the DFT is a real factor. The FFT is just a faster way to compute the DFT.
If we compare the formulas for the CTFS (continuous-time Fourier series) and for the DTFS, we see they are analogous, the only main difference being that the DTFS requires a finite number of harmonics (as opposed to an infinite number as in the CTFS). On the other hand, if we compare the formulas for the CTFS and the DFT, besides the differences between the CTFS and the DTFS, we additionally see there is a scaling factor difference. Thus, the DTFS is more analogous to the CTFS than the DFT (if you disagree on this, please explain why). But why do we use more often the DFT than the DTFS? Equivalently, why was the FFT algorithm built for the DFT instead of the DTFS?