Hm, bear/born turns out to be a tricky one, maybe just in British English (not certain whether borne is a form used in American English).
The Cambridge dictionary lists (be) born as a separate verb. Maybe it's to avoid two past participle forms? e.g.
She was born into a rich family. vs She had borne him two children.
Or maybe it's the point of view.
Ended up googling for I was borne with the born meaning (rather than carried etc), and even found the two forms in the same sentence, with the same basic meaning. It's a translation of some old Hindu text, so it's probably also made to sound archaic even if it's a recent translation.
She was the daughter of a hermit, and was born into the world in this position in consequence of a curse; and I was borne by her to this excellent Brahman...
Maybe borne in this case accentuates on her bearing the child (to this excellent Brahman no less)? On that line, not too sure whether borne would be usable for things that aren't actual procreation. An idea was borne, a new legend was borne ... those don't look good to me, but maybe they'd have been fine in some more archaic text.
As for deponent verbs... I actually had to look up what it means, and pretty much everywhere it came down to Latin or Greek and how to translate them into English, which probably means English doesn't have them.
The definition wikipedia gives is: In linguistics, a deponent verb is a verb that is active in meaning but takes its form from a different voice, most commonly the middle or passive. A deponent verb doesn't have active forms; it can be said to have deposited them (into oblivion).
bear-bore-born doesn't fit that, it has active forms; if be born is to be viewed as a verb on its own it doesn't fit the description either since it's not active in meaning.
