My starting point is the feeling that, although it no doubt diverted Axis resources from the allied or any point of view fighting up a relatively narrow mountainous penninsula is not ideal.
Absolutely - but they did anyways in OTL. That's why your Corsica-strategy would allow a short-cut. (By the way, I read today that a large part of the forces for Dragoon got assembled in Corsica, so the island seems to have been infrastructured well enough to allow such).
My feeling is, that a sort of Dragoon operation (this time only bypassing the French Alps via amphibious operation) could have been accomplished shortly after Overlord anyways. As I read, OTL dragoon was a plan on short notice anyways. Thus, it is highly probable that a different operation in Italy (and by mid-1944, on the Balkans!), wouldn't negatively affect the liberation of France.
Getting into Southern France instead of Southern Italy by 1943 repeats the mistakes in Italy in a different country:
- Allies invade the economically weaker part of France resp. Italy
- Allies invade the part of France resp. Italy which is further away from the German border
- On the way from Southern Italy as well as from the Provence, mountaineous territory has to be overcome in order to push Northwards.
An advantage of the French strategy, though, would be that a lot of Axis reserves would be bound simply by guarding the Italian coast...
To me though, the main question, is whether an attack from the vicinity of Genua northwards is doable at all. I haven't been there yet, but even the segment of the Appenin northwest of Genoa looks very challenging for a modern military advance.
On the other hand, it is only 55km from Genoa to Novi Ligure a city where you are definitely in the Po basin. From there, the main objectives in NW Italy, Torino and Milano, are only 120 resp. 95km away. From Milano to Venice, the strategic objective, it is 265 km. That is altogether 415km of advance, with one major river crossing necessary (though an advance on the Southern side of the Po is a good thing as well) and once the first few dozen km are done, in good terrain.
It is 415km from St. Tropez (in the centre of the Dragoon landings of OTL) to Lyon. And then you only have the control over Southern France.