The answer will be disappointing, I am afraid, but typical, "natural" and "artificial" are very relative notions. The "discovery" of the focus-directrix property was likely a technical lemma in investigations of solid loci, which was later also used in the study of curved mirrors. When it did appear Greeks did not make much of it, its modern prominence in the theory of conic sections is a late artifact.
The earliest extant source we have is Diocles' On Burning Mirrors, and he is working off of Euclid's previous work on spherical mirrors. The point was of course to find a shape that would gather sun rays (assumed to be parallel) at a single point, the focus. As Knorr pointed out, reducing ray-focusing property to the focus-directrix property to "discover" that it is possessed by parabola would require something like solving a differential equation, which was beyond the devices of Greeks. He then speculated that someone (perhaps, Archimedes's friend Dositheus) made a "lucky guess", and then proceeded to show that the curve constructed by the focus-directrix property is a parabola (similar demonstration for ellipse is described by Anthemius). Acerbi suggests a different reconstruction in The Geometry of Burning Mirrors in Greek Antiquity, where one can find geometric details of how the property came up as a lemma:
"Focal properties of the parabola were obtained as a by-product of an attempt at proving that the property of the subtangent is a sumptoma of the curve. About the relevance of this property to the eyes of the ancient geometers it is not necessary to spend so many words. Recall in fact that the property of the subtangent was well known to Archimedes... As a consequence, the focal property proves crucial to employ the subtangent as a method to construct a parabola with given vertex and parameter (cf. Con. I.52, that, however, does not lead in an immediate way to a pointwise construction) and makes it possible to solve by reduction a particular case of the problem of tracing a conic section after some of its tangents are known.
"[Assumption] is that a property so important (to us) as the focus-directrix could not be reduced to the status of mere functionality to a pointwise determination of a parabola, as indisputably happens in Diocles’ approach and as Knorr himself has to recognize... Of this property, there is no trace in Apollonius’ Conica as they are now extant (recall that it holds in reference to the axis only, and therefore it is not a sumptoma that can be referred to any diameter).
"Toomer surmised that Diocles himself was the discoverer of the property in the case of the parabola, its validity being extended after him to the other
conic sections. The few lemmas Pappus offers, at the very end of book VII of his
Collectio... is the only attestation in the ancient corpus of the fact that the focus–directrix property is a sumptoma of the several conic sections (the formulation as a locus bears this out exactly).
Zeuthen (1886, pp. 210–215 and 367–371) observes that, if Pappus had to make such a lemma explicit, then the focus–directrix property should have been taken for granted in the Loci on a Surface. Zeuthen’s remark can be completed
by the conjecture that the most “natural” place where the property was established, of course in the above form of a sumptoma formulated as a constraint identifying a locus, is in Aristaeus’ Solid Loci... In sum, if one is not willing to venture in empty conjecturing, then a minimal hypothesis is that the property had been employed in the Loci on a Surface as an intermediate step in the analysis of some locus, but that it was simply considered as a sumptoma of a conic section".