"Burgle" is not a particularly common verb (746k results for "burgle" with quotes on Google) compared to "burglar" (40m), so it's not on the tip of the tongue.
The American instinct is then to reconstruct a verb from the more common noun, and the -ize pattern fits.
I'd guess that if you asked an American, they'd say that a burglar robs.
I know online dictionaries say both are correct, but irregardless[1] it's one of those words that really irritates me.
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[1] Yes, using that word is an attempt at written humour.