Why is the word "plait" pronounced "plat" when it looks like it ought to be pronounced "plate"?

Because it’s french. The same people who spell Oui and pronounce it “we”

hold on while I plate my hair

Because the French enjoy words that can be said with a sneer while waving a cigarette in the air dismissively, and it's easier to scoff at a PLAT (hell, if you say it right, it's like spitting) than admire a plaaaayyytehhh

The astronomer Phil Plait pronounces his last name “plate.” The actor Oliver Platt pronounces his name “plat.”

French origin, I think.

The various apparent irregularities in English arise from it having formed from many languages from Roman times (Latin) onwards: Greek, Saxon and other Germanic, Scandinavian, French, Italian, Asian, etc.

We Britons also respect etymology despite linguistic evolution, so for example know the difference between the roots [i]paed[/i] and [i]ped[/i], and the proper spellings of the Metric System / ISO units invented in France.

.
As for the title of this Group ("I WTF") does not that show just how shallow this site can be, if it invents ones like that?

Just to piss you off, really.

Maybe for the same reason "plaid" is pronounced "plad."

A French word hence French pronounciation.

English. spread across it's empire, changed a lot, but in some places stayed the same.
consider ass. arse and ahse

English is full of such things. I am told that many modern pronunciations, or once much closer to what the spelling looked like

From the verb plaire