Comparing the Fourier transforms of various instruments is indeed the right idea, as a difference in the amplitudes of various harmonics does lead to a difference in timbre. So the answer to the question, "Why does the theremin sound odd?" is in some way answered by the Fourier spectrum of its sound. But the answer is a bit more complicated than "it has most of its power in the lower harmonics." For example, the french horn at moderate volume also has most of its amplitudes in the lower harmonics, which gives it the characteristically sweet, pure tone that it's known for. But we don't think of the french horn as sounding artificial.

So what makes us think of something as artifical-sounding? The answer is most likely just that we haven't heard an instrument with a similar set of harmonic amplitudes before. The definition of "similar" is quite complex, as our ears are characteristically more sensitive to higher frequencies, and so the particular arrangement of the upper-harmonic amplitudes plays an inflated role in the determination of the timbre. But the point is that it has very little to do with the actual design of the instrument itself, as instruments with very different designs can have very similar Fourier spectra, and thus, very similar timbres.

One point that I would like to make clear is that it doesn't have much to do with the theremin being electronic. As an example, here's another instrument that is played in a completely classical way that sounds just as artificial as the theremin: https://www.youtube.com/watch?v=i5KiodkF2m4. The instrument is known as the "musical saw," and is literally just a carpenter's saw being played with a violin bow. If you grew up listening to the singing-saw music in Dmitri Shostakovich's opera "The Nose," when you first heard the theremin, it wouldn't sound odd or artificial at all - it would sound just like the singing saw you were used to!

So the precise reason for an instrument sounding artificial is not really a function of any particular aspect of its design; rather, it's a function of you personally being used to a particular repertoire of Fourier spectra. When an instrument comes along that, for whatever reason, has a substantially different Fourier spectrum, then you'll perceive it as sounding artificial. Answering the question, "What particular instrument designs produce Fourier spectra that are substantially different than what we're used to?" isn't really possible, as there are an uncountable variety of different designs, and even an uncountable variety of _types_ of different instruments you could conceivably make. For example, if you took a cavity of exactly the right shape and blew some pressure-modulated air into it, you could make a wind instrument that would sound exactly like the theremin.