If it's mandatory it's mandatory. In France winter tyres are not mandatory but chains are (on certain roads). Don't ask me why.

Secondly yes, You need them in the mountains from time to time. Driving from a tight hairpin in a steep uphill with FWD can make You stuck very easily. Also don't forget that there is a lot of traffic on these roads which means that plenty of times what You could get through, if You were alone there, is undriveable because the traffic is too slow or because You must stop in uphills because of others. That's why in all Alpine countries chains are mandatory on some roads (in Austria they use digital traffic signs and activate them only when the weather requires so). Don't know how is that in France but in Austria if You have 4x4 You can go without chains I think.