I had my GTI in the Alps at Christmas, and it was also simply dreadful in the snow and ice. Turning ESP or starting in higher gear did not work. Resorted to using snow chains, which are a major pain to get on, and especially off. After the last removal, I had no ESP anyway, as I'd damaged the wiring to the ABS sensor on one wheel which cost a lot to get fixed when I got back.
The solution is winter tyres and/or 4x4 with winter tyres. The regular tyres (mine anyway, Contis) have no lateral tread.
But to maintain the thread, I was surprised how bad it was, I knew rear-wheel drive cars would be bad (and they were) but the GTI was no better in practice.