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Yeah, it only takes one cold corner. I've been in a bunch that's hit totally invisible black ice on a sunny corner and done collarbones. It must've been minutes away from being water rather than ice. Even standing on it to pick people and bikes up was difficult and yet it looked like normal road. I've gone back to riding 25mm tyres though for this year's production of Hippy on Ice.
When there's ice around I normally lean, then go "oh shit" and totally readjust mid-corner. Maybe grab a handful of brake just to seal the deal. Surefire way of hitting the deck. Ice is for drinks, not roads.