Alexander in late WWII wanted to do a major push from Italy through Austria, but wasn't supported. Alanbrooke considered it too difficult (and he didn't respect Alexander very much, considered him stupid!) and anyway the Americans wouldn't be likely to support anything giving a British general the coup de grace. But had it succeeded it obviously would have provided the allies with a chance to go for the heart of the axis and even be in advance of the Soviets.
The terrain between Italy and Austria isn't entirely mountainious, and has provided many historic battlefields. I wouldn't exclude a success, but I'm in doubt if the chance of success is 5, 50 or 75%.
The OTL Italian campaign was the only realistic option to the allies in 1943 and thus provided an option to engage a part of the German Wehrmacht. They certainly had hoped for a quicker advance, not at least by flank landings, but the allied doctrine simply wasn't opportunistic enough. It was discussed if the troops engaged in Italy should be deployed elsewhere, but not at least Alanbrooke pointed out that disengaging the units would only mean many months of units moving and not engaging the enemy. So most units were left in Italy and newly raised units were appointed and trained for Overlord etc.
Churchill constantly fantasised about intervening on the Balkans, but by 1943 I doubt the allies had the numbers, logistics or training to engage the Germans on such a wide front as on the Balkans (or France). Landing on the Balkans in 1944 instead of France would only have brought the allies further away from the German heartland and seriously risk meeting the Soviet flank while the Reds advance on Paris.
In that context I think the Italian campaign was best possible utilisation of allied resources. I do wonder however, if the Italians in 1943, would have been able to throw the Germans out - something like what the Finns did in 1944. If the allies had been at the Alps in late 1943 the Germans indeed would have been in a even more difficult situation.
Regards
Steffen Redbeard