I think it's much easier and more direct to visualize the time-domain as superposition of helical components and the transform as an exploration of what happens when you twist the "cylinder" with varying "intensities".

That doesn't sound very clear at all to me.

You avoid the vague center-of-mass spike depicted here and start from the get-go with the terms of the transform.

The center-of-mass spike is the result of summing across all the different complex points/vectors, this is stated very clearly by the FT formula (sum (int_^) of points on a circle (exp(t)) amplified by signal strength (f(t))). Seems very explicit to me.