Well for a start - they have no need to surrender. ...
Thats fine, the Italian economy was collapsing rapidly. Mussolini can keep Italy in the war, but weapons production was stalled, morale failing, coal & food shortages rampant, rationing system had broken down. Odds are the Facist Grand Council will take action on making peace before the end of the year.
... Which means German troops aren't deployed to Italy, they are free to be used elsewhere. ...
Withdrawing the two armies they had in Italy does leave it open to a easy invasion later.
... It also means, the Foggia airfields aren't available to the Allied Air Forces to attack targets in Southern Germany, Austria, and Romania. ...
In 60 days of capturing Corsica the US AAF & the French had 500+ bombers based there & over 600 interceptors and reconissance aircraft. More capacity was set up on Sardinia. The Allies had the ability to expand the airfields on those islands as a substitute for those further south.
... Unless, you plan to have D-Day much earlier, it means the landings will be quickly overwhelmed.
Why would this landing be overwhelmed? The Brits had 12+ combat experienced divisions available in the Mediterranean. The US had another half dozen, and there were eight French divisions being rearmed. The Germans can try to throw everything at that, but it will be against a Allied airforce that out numbers theirs nearly 3-1 in mid 1943, and it would leave their defenses elsewhere weak with no reserves.
... (are the port facilities in the Western Mediterranean and southern France capable of supporting a force similar in size to Overlord?), ...
The southern port group; Marsailles, Toulon, Nece, Cannes, ect... had a nominal peace time discharge capacity of 18,000 to 25,000 tons daily. Depends on whos numbers you use. Using the lower number & the logistics guide for the Overlord planning of 900 tons daily per attacking division in a army, then these ports can support a army group of 20 divisions, plus army & corps overhead and a tactical airforce ashore. Thats assuming the Allies dont expand capacity. OTL the port of Cherbourg with a peacetime average of 8,000 to 10,000 tons daily was expanded to 20,000 tons daily at 45 days after it was captured.
My notes show the cross beach supply intake of the Rivera landing sites was 5,000 tons daily during the first couple weeks, but that seems low since when Marsailles/Toulon were captured the 6th Army Group had eight divisions, two Army HQ, three Cops HQ, and several tactical air groups ashore.
I think that this would likely have resulted in an Anzio x10 in Southern France. ...
Not entirely a bad thing. Kesselring was forced to cease the counter attacks against the Anzio beachead as the Germans were losing men and material in those faster than they could replace them. Ditto for the battle further south around Monte Cassio & the Arrunci Mountains. The German replacement problem was aggravated by Allied attacks on their transport system. The first 53 days in Normandy resembled Anzio, with the Allied confined to a lodgement far smaller than they planned. But, in the end the Germans lost over 440,000 men in the Normandy battle vs the Allies 225,000