I was sort of excited about this because it seemed like they might be suggesting there was actually a window here to explore a quantum theory of gravity. Alas, they say "Although the methods presented here suffice to describe the entanglement of clocks arising from gravitational interaction, a full description of the physics with no background space–time would require a fundamental quantum theory of gravity".
To your specific question "Who's to say that the inaccuracies are merely caused by fluctuations in spacetime?" I don't think they're claiming they found a theoretical cause to explain some observed fluctuations. In fact, they put in some notional values to see what comes out and observe "this effect is not large enough to be measured with the current experimental capabilities".
Now, I think what you're getting at is the idea that you would expect the ticking of clocks to vary over space time. What they're saying is, "Of course, but we believe the problem is even more fiendish: the tick rate not only varies but it accumulates uncertainty to the point that a system which starts out as a clock can no longer be considered a clock at all. Further, [what I believe they are saying] the clock's coherence decays in a way that potentially varies with path and frame of reference. At which point, one starts wondering, why haven't we all flown apart already?
And they're doing this, again, all from a theoretical standpoint. It' the same sort of thought experiment you might get if you modify Euclid's fifth axiom: all sorts of weird stuff is possible, and we don't really need experiments to show the math, although there are experiments that can be done.
(1) http://www.pnas.org/content/early/2017/03/06/1616427114.full...
Who's to say that the inaccuracies are merely caused by fluctuations in spacetime? If you turn on the radio to a frequency on which there is no known transmitter, you hear background radiation. Is it not the same for spacetime? As you measure time (space) more precisely (more instantaneously) and more often, your measurements will occur more often on various boundaries of the background radiation of gravity (which we all know fluctuates spacetime). At that point, the is the error rate you see not also the rate of background radiation? If you could measure or predict the velocity of multiple background spacetime radation sources, you could reduce error? Thus, the error rate of measuring time is the rate of change of gravity.
Of course we built gravity wave detectors, so that nullifies that whole thought. Right? Or does it? The collision and merging of celestial bodies is arguably one of the biggest events in the universe. But what if spacetime or gravity (are they the same thing? I don't know) 'bounces' back on like a wave crashing against a wall, then surely there's fluctuations in spacetime due to past events. Wouldn't that intrinsically cause minute "errors" in measurements?
What keywords should I search to find (hopefully free) online resources to answer my questions?