I before E except after C

What actually happened

Start with the exceptions and the rule dissolves: their, weird, foreign, science, seize, height, protein, neighbour, sufficient, ancient, efficient, glacier, either, neither, leisure, forfeit, heifer, caffeine, beige, feisty, and society, which manages to break both halves in one word. Note that science, sufficient, and ancient are exactly the ei-after-c cases the rule promises, and they go the other way.

The 2009 guidance in the Support for Spelling document advised that the rule was not worth teaching because it is unreliable and hard to qualify. That produced predictable outrage about political correctness, which had nothing to do with it. The complaint was that the rhyme is a bad rule.

There is a version that works better: the rule holds reasonably well when the letters spell the "ee" sound, as in believe, chief, and receive, which is what the fuller rhyme's "or when sounded as A, as in neighbour and weigh" was fumbling toward. That is a defensible teaching point. The short version drilled into most of us is not, and English spelling is irregular enough that a rule with a hundred exceptions is just a hundred exceptions.