I think you are right, and it seems the frequency domain aspect could also have won for historical reasons: the first "SAR" systems were effectively performing

I think you are right, and it seems the frequency domain aspect could also have won for historical reasons: the first "SAR" systems were effectively performing doppler beam sharpening by passing the output of a conventional radar through a filter bank and mapping the output of each filter to an off-track angle. But again, when I first got introduced to (modern) SAR I found the explanation through Doppler shifts to be more confusing and less physically understandable than a purely geometric approach