I've always understood tempering as a process to enable something to be more easily molded into something useful--to make it more malleable and robust--in this case the Fourier transform. As nLab puts it: The main property is that the Fourier transform of a TD is well-defined, and is itself a TD; and that it naturally extends the standard FT. This makes TDs the natural setting for solving (linear) PDEs.
Consistently, in metallurgy, tempering increases the ductility of a material, i.e., "the degree to which a material can sustain plastic deformation under tensile stress before failure," making it more useful.
Similarly, to temper one's emotions is to guide them, shape them, mold them, into productive channels, or at least less destructive / disruptive ones.
'To temper' has a long history. From Oxford Languages:
In Latin, temparare--to restrain, moderate $\to$ Old English, temprian, and Old French, temprer--to temper, moderate.
Old English temprian ‘bring something into the required condition by mixing it with something else’, from Latin temperare ‘mingle, restrain’. Sense development was probably influenced by Old French temprer ‘to temper, moderate’. The noun originally denoted a proportionate mixture of elements or qualities, also the combination of the four bodily humors, believed in medieval times to be the basis of temperament, hence temper (sense 1 of the noun) (late Middle English).
(I'm pretty sure Schwarz understood French.)
We use, in modern English, 'a temperate climate' to mean a moderate climate between tropical and harsh northern climates--a comfortable mixture of the two. We can say, "The climate of the coastal regions is moderated by the cool waters of the Pacific," but never, "The climate of ... is temperated by ...," rather, "The climate of ... is tempered by ... ." Temperate is purely an adjective like 'mild' whereas tempered is a verb (past participle) used as an adjective meaning to have been tempered--same grammatically as burned in 'a burned / burnt car'. Even the pronunciation of moderate changes according to whether it is being used as an adjective or as a verb or the past participle adjective moderated.
(Maybe a native French speaker can comment on parallels in the grammar in French.)
From all the considerations above, I see the delta 'function' as an construct of Heaviside and Dirac that has been molded into one amenable to the tastes of purist mathematicians (It's NOT a function!--the shrill mantra)--a morphing not really necessary for pragmatic physicists and engineers--by Schwarz and his theory of distributions. In that sense the delta function as a distribution that has been tempered by Schwarz and his theory seems fitting, just as the climate of coastal southern California is tempered or moderated by the Pacific waters. Of course one is free to say the climate along the coast is temperate, but that doesn't stress an agent and action that results in that quality.
To end with the Bard:
1591, William Shakespeare, Henry VI, Part 1, Act II, Scene 4, :
Between two blades, which bears the better temper: […]
I have perhaps some shallow spirit of judgement;
But in these nice sharp quillets of the law,
Good faith, I am no wiser than a daw.