What then of the commonly heard "brang" and "brung"? How would this fit into this pattern?

The model for such formations is sing-sang-sung, drink-drank-drunk[en], etc. There are enough such "correct" conjugations among frequently used verbs for "incorrect" ones to creep in by analogy. It helps that bring-brought-brought is itself irregular and thus hard to remember.

Sometimes such a construction survives long enough to become "correct". For example, the past tense of "dive" used to be "dived" everywhere, but in US English it's now mostly been supplanted by "dove".

A much older example is wear-wore-worn, which was a weak verb in Old English and would regularly have developed into wear-wear-weared.

Perhaps there is some intuitive understanding of how strong verbs conjugate that is then misapplied in such cases as this?

It's the same kind of analogical reasoning that makes strong verbs become weak. If the pattern is common enough and obvious enough, it has a good chance of being reproduced.