They would have had to go around Switzerland but that seems like a better plan than storming a beach with no cover.
There were many political and military considerations for an Allied beachhead on the Atlantic wall despite the foothold in Italy. Perhaps the single-most important factor leading the Allies to undertake Operation Overlord (the invasion of Normandy) was political - Joseph Stalin and Franklin Roosevelt demanded that the Western Allies open up a western front to relieve pressure from the Soviets. But the military benefits of a French landing given the topography of Italy should not be ignored either.
By 1942-43 the course of the war had decisively turned against Germany, with setbacks on all fronts: in the East, Friedrich Paulus's 6th Army was decisively defeated at Stalingrad; to the South, Axis had been pushed out of Africa and the invasion of Italy was underway; in the Atlantic, the U-Boat threat was ending due to improved Allied convey escorts, submarine detection methods, and air cover; and at home, the combined might of the Royal Air Force and the US Army Air Force had begun the large-scale bombing of German cities.
Nevertheless, the Italian campaign was not progressing with the speed that the Allies had hoped for. Although Italian resistance was mostly token, the Germans rushed two extra divisions (29th Panzergrenadier and the 1st Parachute) to Sicily to join two divisions already in place (15th Panzergrenadier and the Hermann Goering), under the leadership of Field Marshall Albert Kesselring, who had successfully stalled the Allies in Tunisia for six months. There, the skilled resistance of front-line German troops, using the natural topography of Mount Etna and the Messina Peninsula, bogged down Patton and Montgomery for weeks. "Kesselring, with sixty thousand troops, had held back two Allied armies for thirty-eight days, inflicting twenty thousand casualties at a cost of twelve thousand" (Smith, Eisenhower, 281). And in the end, all four German divisions were able to evacuate intact to mainland Italy, along with 70,000 Italian troops.
The invasion of mainland Italy was no easier. The combined debacles of Sicily and the Allied bombing of Rome let to the ouster of Mussolini by King Victor Emmanuel, who replaced him with Marshal Badoglio. Although Badoglio entered in negotiations with the Allies, the German army swiftly disarmed their Italian counterparts and occupied Rome. The Allied landing at Salerno (near Naples) was almost immediately repulsed by stiff German resistance. As they had done in Sicily, using the natural terrain of the Italian peninsula, the Germans dug in and stalled Allied advances. Looking at a topographical map of Italy should prove a clear answer to your question, at least from a military perspective: Italy is mountainous, narrow, and filled with rivers that run East-West. These factors made it a nightmare for the Allies to advance quickly, and made it possible for a smaller number of German defenders to hold out against numerical superiority. Even if the Allies had somehow managed to push through the majority of the Italian peninsula, they would reach the Alps of Austria and Savoy, where the Germans could mount yet another mountain defense.
The military benefits of a channel invasion are clear and numerous. There were many quality airfields available in Britain to provide sufficient air support for troop transports crossing the channel, and to aid in an amphibious assault, and to delay German reinforcements to the coastal defenses. Britain could provide many more divisions to the operation: if the invasion was limited to Italy (or had a Japan-first strategy been undertaken), home isle security would limit the amount of soldiers Britain could provide. France is more open compared to the rugged terrain of peninsular Italy, allowing for the Allies to push a larger front simultaneously. Northern France is also in closer striking distance to German territory compared to much of Italy.
From a political standpoint, both Roosevelt and Stalin were in favor of a Normandy landing. Churchill was less sanguine, and favored a variety of expeditions, including an invasion of Rhodes, which he thought might force Turkey to join the Allies. By the end of the Tehran Conference in 1943, the Allies agreed amongst themselves to open the Western front. And in concert with their Western Allies, "the Russians launched an offensive in the east to coincide with the Normandy landings" to further strain the resources of the German military (Fulbrook, 196). Plus, the Allies could take advantage of the animosity of the French towards the German occupiers in Northern France, hopefully undermining any determined German defense like in Italy.
To summarize, the Italian front was progressing very slowly, and there was little reason to be optimistic about it. The French front forced Germany to commit an even greater number of resources to reinforce the Allied breakthrough of the Atlantic Wall, drawing desperately-needed forces away from the massive Eastern Front. It met Stalin's demands, thus maintaining a working relationship between the Western Allies and the Soviets. A map comparing Italian and French geography should reinforce the decision that France would be a better target for the Allied invasion of Europe, when the goal is to capitulate Germany as quickly as possible.
Sources
Fulbrook, Mary. A Concise History of Germany. New York: Cambridge University Press, 2013.
Smith, Jean Edward. Eisenhower in War and Peace. New York: Random House, 2012.
Smith, Jean Edward. FDR. New York: Random House, 2007.