Smaller villages are more conservative in their dress. Being dressed like you are at the beach may not be seen as appropriate and certainly not if you intend to visit churches anywhere in Italy. Restaurants, as opposed to cafes, might also frown on brief clothes.
White capris or similar would be fine but a hassle to keep clean.
If you are concerned re being cool, then dresses of natural fibres such as linen or cotton are actually cooler and more absorbent than shorts.
Capris would better serve than shorts, and white isn't practical or the best option for travel.