Yes. The problem is assuming the only thing that matters with the fourier transform is the DFT. Point #1 is incorrect. Sometimes sampling theorems are misleading (not incorrect). There are cases where it is highly advantageous compute a continuous fourier transform without a DFT.
I think my answer was highly oriented towards handling real data. But even in simulation, any calculations are discrete.
It helps tremendously to understand continuous Fourier transforms, because most of the theorems have immediate discrete analogs.
