Does it make sense to express a given FT from 0 to 1 as a convolution?
As for the why, it's always good to come up with alternatives, even if one doesn't immediately see their benefit. In this case it's about efficiency, especially if the FFT length is a prime. The resulting algorithm can also be used to solve the more general problem of computing (samples of) the $\mathcal{Z}$-transform on circles or spirals in the complex plane. You can also zoom into a certain frequency region without the need to compute the DFT over the whole range.
The resulting algorithm is called Bluestein's algorithm or Chirp Z-transform. The basic trick to represent the DFT (or $\mathcal{Z}$-transform) by a convolution is to write
$$nk=\frac{n^2+k^2-(n-k)^2}{2}\tag{1}$$
and use this expression in the original formula. This results in the representation of the original DFT sum by $(i)$ a multiplication of the original sequence with a chirp sequence, $(ii)$ a convolution, and $(iii)$ another multiplication with a chirp sequence. The details are explained in the link cited above.