Note that although there is a bigger drop from saddle to stem on Contador’s bike the default modern position in the hoods is actually higher then the old in the drops position of Coppi.
I plugged in the geometry of Coppi’s 1949 Bianchi which can be found in the Winter 2008 issue of BQ and a Specialized Tarmac SL3. It doesn’t look it in the photo but the front and back are equally shorter so weight distribution does not change as was pointed out in the comments. I thought the longer stem on the modern bike might provide additional leverage but Coppi’s stem was 115 and I chose a 130 for the Specialized and looking at the drawing the difference is not dramatic. For a rider switching between the hoods and the drops while racing the contact points are very similar between the 2 geometries.