Why does "stigmata" [often] have penult stress?

Generally in English the addition of an inflectional ending does not affect stress. That is, the ending itself is not stressed, and stress remains on the same syllable as it is in the uninflected form. For instance, even though in morphologically simple forms, we don't get 3 unstressed syllables at the end of a word in English, the addition of the -ing inflection of the progressive aspect does not affect the stress, even when it is added to a verb ending with 2 unstressed syllables: interest/interesting, amnesty/amnestying.

Likewise, the regular English plural ending is inflectional, and its addition never causes stress to shift to the right, nearer the end of the word: parentage/parentages, Cavendish/Cavendishes.

Therefore, if the -ta is a plural ending and is counted as an inflectional ending, like the regular plural ending, one would predict that stress would be unaffected by adding it, and you would get stigmata ending in 2 unstressed syllables. That is a big "if", though, since I don't think stigmata is actually the plural of stigma in current English, and I never heard it pronounced on anything but the penult. So I think the stress you are asking about is archaic in English.

This is a beautiful question that should have garnered a few hundred upvotes by now.

If I may venture a theory (since no official theory seems to exist at this point).

"For example, do English speakers have some natural preference for a penultimate stress on polysyllabic words?"

You may be onto something here. Speakers of languages that lend themselves to versification ... let me rephrase this: ... that are conducive to composing rhyming poetry ... poetry that rhymes ... have an affinity for ... well, rhyming.

The same people's fondness of meter is less pronounced, but isn't far behind. We're all poets in our hearts.

Today, when professional poetry is hibernation, everyday, mundane, if you will, rhyming is alive and well in everyday, mundane speech. Itsy-Bitsy spider and all.

If your boss' name is Matt, he'll never be just Matt: he's Matt the Rat. Coppers always gorge on Whoppers. A particularly intelligent woman is Wits with Tits. The CEO of a think tank is Main Brain. And so forth.

Now STIG-ma-ta, with the stress on the first syllable, rhymes with absolutely nothing at all, whilst stig-MA-ta rhymes with sonata, cantata, regatta, persona non grata, and terra cotta.

In conclusion I'd like to point out that in this our epoch of habitual unmitigated hypocrisy and crass doubletalk, the truth oftentimes strikes people as comical.