iZotope RX's FFT processing?
Spectral Repair is more complex than just muting a handful of harmonics.

But first, let's just clarify that as far as Linear Time Invariant (LTI) systems are concerned, their processing can be applied, equivalently, either in the time domain (via the operation of convolution) or the frequency domain via either the overlap-add or overlap-save methods.

iZotope's Spectral Repair performs interpolation. That is, to remove an unwanted portion of a recording, it will try to "mimic" the characteristics of the spectrum in either ends of the unwanted region.

Consider the following image:



Can't we just remove the railings by copying and pasting the grass from another (and very similar) part of the image and adjust its brightness? Couldn't we even replicate how the green colour varies in the grass portions and just generate some "new" grass for the regions of the railings?

Is it going to look odd? Possibly, if it was overdone, but for small regions it may be good enough to fool the eye.

This is what the "Replace" mode (and "Attenuate") does in iZotope, but instead of grass and railing there is background noise and local disturbance.

It's not so much attenuation (or "setting harmonics to zero") as "masking" or "hiding" the unwanted sound, since it attempts to make up a good enough patch of harmonics to bury it in by "looking at" the surroundings of the disturbance.

For more information on interpolation please see this link. For an example of "learning the profile and applying a filter" please see this link and this link.

Hope this helps.