In over 15 years of ski trips driving between Geneva and various ski stations in the Alps, you are over thinking this. We’ve had all sorts of road conditions from sheet ice to deep snow and never struggled getting where we needed to be. If the forecast is bad, just plan to allow some extra time for your journey particularly on the way home when you have a flight to catch, but generally, the French really are efficient at keeping things moving.
We always have hired from the Swiss side of the airport. Cars always come fully equipped with winter tyres and chains. Only ever had to use the chains once. But as chains are a total pain to put on if you’ve never done it before, particularly in a blizzard and when its below freezing, always best to learn how to do it somewhere warmer first. Snow socks are a far easier method to get you out of trouble than chains, and yes they really do work well.
We generally break up the journey for supermarket shopping somewhere around Annecy. Also there are lots of roadside bakeries at various roundabouts between Geneva and the French Alps. Some of the cakes are out of this world.