I think the answer depends on what you mean by the term "half cadence."
On the one hand, "half cadence" in most theory books today is basically defined as a move to the local dominant. Your first cadence clearly can't be a "half cadence" in that sense, since it's about the strongest IAC one could imagine (and the soprano even jumps down to make it seem like it just barely missed being a PAC). There's absolutely nothing in that first cadence progression that remotely could be classified as a stereotypical "half cadence" to a local dominant.
On the other hand, I sense that you're after something different. The etymological root of half cadence obviously is referring to a sort of standing point on the way to something else (i.e., "halfway" to there, whatever "there" is).
And in that second sense, one might consider a lot of chords as substituting to create a sense of an "unfinished cadence" that ultimately gets to a PAC later.
I certainly think III can function that way in minor keys. The immediate progression that comes to mind is the standard Folia progression and variants of it, where phrases commonly move toward III (the mediant) but then return to minor. Hundreds of Russian Orthodox chant arrangements do a similar progression too, mirroring a minor-mode harmonic move that was also common in Western Europe in congregational four-voice chanting settings in earlier days. (This is the tradition the hymn setting you quote ultimately comes out of.)
In fact, early theories of keys and cadences in the 17th-century often gave a place of privilege to cadences on the mediant, as another sort of "dominant" that served in contrast to the tonic. This relationship was usually stronger in the minor mode.
(Actually, to be more accurate: originally the French terms dominante and mediante were used to classify cadences that could occur on various notes of the scale depending on the precise mode, where the dominante was only slightly inferior to the finale of the mode, and the mediante was an alternative place to cadence in the middle of a piece, which often fell on the third above or below the finale. A cadence mediante was actually sometimes a term for any cadence that occurred "in the middle," i.e., what you're calling a "half cadence." The mediante cadences often came on the third above in what we'd now think of as "minor modes," but could fall below in some modes we'd now think of as "major modes," i.e., a cadence to vi. Eventually, the terms mediante and dominante settled down in French theory to the third and fifth degrees in all modes, which begat our modern nomenclature for those scale degrees.)
So yes, there's all sorts of conceptual and historical precedent for saying that III in a minor key can be an alternative cadential destination for a medial cadence within a larger set of phrases.
All of that said, I don't understand the labeling of cadences in the question. Both of the cadences to G are obviously authentic cadences with very strong ii6-V-I motion. I get that the first cadence feels less strong (as an IAC, lacking a V7, and a very fast modulation), but it's clearly not a half cadence in the hierarchy where cadences are called things like PACs.
I think perhaps what you might be feeling is partly the general sense of 4-bar phrase units adding up to a larger 8-bar structure. When one hears a sudden modulation at the outset and an arrival point after 4 bars, it's often meant to feel like "halfway there," and you get the rest of the way "there" after the first 8 bars. That seems to be the sort of modulating period-like structure you document at the end of the question, giving a sensation of structural completion after 8 bars.
And, in this particular hymn, by the end of the first 8 bars, one might legitimately wonder what key we're even in. Did we just begin on an unusual chord, and the "real key" is G major? Of course, subsequent phrases eventually confirm E minor to be the true tonic (after even more waffling with G -- the mediant -- as a temporary goal again toward the end of the verse).