The closest to Geneva you are the easier it is to just do Mont Blanc as it's almost all highway except for the two bits up and down the tunnel. The other ways will take more time.
Where in Vaud are you? Would say it makes a different if you are in Nyon or much further from GVA.
We always go by St Bernard to Aosta. In Summer, if we have time, we will take the pass- but in winter it has to be the Tunnel. Drive up to Martigny and up and over- easy peasy.
If you search for the route on Google via Itinéraire Michelin- it will give you all the instructions and prices, tolls, etc.
Just done a search, and they don't advocate St Bernard actually
https://fr.viamichelin.ch/web/Itiner...ewEngine=false
just click on a route and they will give you the full details.