I've heard people say that the reason US in WWII preferred designs with fabric control surfaces with either a metal or plywood frame is they can sustain some small arms fire without disintegrating. Is this really the reason? Or is it simply weight saving?
It was done to achieve the most flutter resistant surface for the least weight. To prevent flutter you balance the surface to put the C of G of the surface at or ahead of the hinge line so it can't oscillate in sympathy with the wing, which adds energy back into the wing's oscillations until it all falls apart. To do this you add weights forward of the hinge somewhere. (Hold a piece of plywood by the edge in your hand and wave your hand back and forth. The surface wants to "trail" behind your hand movement because it's center of gravity is away from where you're holding it. If you were able to hold the piece near the middle at its center of mass, it won't do that).
A metal skinned surface is heavier than a fabric one, and therefore requires a heavier mass balance, so the overall weight gain is substantial because adding weight aft of the hinge requires a balancing increase in front of the hinge. But later on as the speed got up there, the fabric covering itself became a problem and designers just went with metal and lived with the higher weight.
A metal control surface can also sustain some small arms fire without disintegrating -- aircraft with metal rudders and elevators used to routinely come home with bullet holes in those surfaces.
The real reasons wood and fabric surfaces might have been preferred (though they weren't always; several designs had all-metal control surfaces) are the classic reasons for everything in aviation: cost and weight. In the late 1930s, when many of the aircraft used in WWII were designed, it was still possible to make a wood and fabric structure, especially a control surface (as opposed to a wing), that would give the needed performance and strength and come in lighter than an all metal one, and frequently at lower cost in volume manufacturing.
As performance specifications continually crept up (higher speed, higher loading, more vibration from a larger engine, etc.), metal surfaces gained the advantage. They could ultimately be stronger than wood and fabric, hold their surface shape over a broader speed and pressure envelope, and as noted, wouldn't disintegrate if they took a few bullets -- in fact, they were usually more durable than wood and fabric, because metal is less prone to tears and splintering.
Fabric covered control surfaces on otherwise all metal aircraft were rarely wood, but rather metal, and it was done to reduce weight, which is particularly important when it comes to the control surfaces, as heavier control surfaces translate into heavier stick movements. Here is the rudder from a Harvard I trainer, which was fairly typical for any control surface - the ribs are stamped aluminium sheet, and the structure was riveted together, and then covered in fabric which is sewn to the ribs. Earlier types often used steel tubing, but that had become fairly rare on military aircraft in WW2, aside from some smaller trainers.
@MeatballPrincess welcome to Aviation StackExchange, and interesting question. I don't feel the answers so far are completely inclusive of all the factors, so I'll include my interpretation of the answer. Firstly, some other variables you could keep in mind was the US's manufacturing capability during WW2 compared to countries that fielded many wood-skinned aircraft, such as the RAF. I'm not completely sure where you got your information that the US preferred wood and fabric over metal during WW2, as we had few shortages of metal and as I will explain, the performance and application of these fighters simply negated any real reason to use anything but. However, metal is expensive, and the UK's situation was too urgent to take the money and time to develop newer metal skinned prototypes. If I recall correctly balsa wood was the main skin and chemically treated stressed fabric was the most common fabric used in aircraft skinning. Of course, planes such as the Spitfire and others are exceptions, but we see fabric and metal making up a large portion of the skin on the most numerous and resource intensive applications i.e. the Hurricane and especially the Mosquito.
To address your point on durability under small arms fire, you could probably use common sense to identify the fire hazard of wood, but again it was treated to be fire-resistant (obviously) and the two materials (aluminum vs wood) have various pros and cons. As a skin, these materials just cover the frame and slightly protect the interior, of which the more vital component will be further shielded with steel plating. When bullets or cannon shells blow holes in the skin, metals such as duralumin retain some measure of rigidity and unless those shells were loaded with some seriously hot chemicals, won't burn. Wood and fabric on the other hand can be easily set off by competent incendiary rounds and will continue to burn away the aircraft's aeronautical properties. The metal used by the US was mainly duralumin, which was an aluminum-copper-magnesium alloy with a density of 2.78 g/cm^3, compared to balsa wood which is 0.16 g/cm^3. Obviously, wood and fabric (which is even less dense) are lighter than metal. However, duralumin is simply just better for fighter aircraft, or any machine with combat application. It resists crumpling and wrinkling (a huge issue for fighter planes back then) to hundreds of miles per hour greater than fabric ever will, and wood has issues two-fold: it can't be easily used in naval theaters as water rots even treated wood faster than it rusts metal, and it burns. As a result we see that the US had pretty much 0 non-metal frame and skin fighters, bombers, or reconnaisance aircraft stationed on aircraft carriers or anywhere for that matter. John K already answered the aerodynamic reasons for choosing metal, but I don't think any of the other answers explains why the US actually didn't use wood. It was duralumin for almost all of their fighters from the onset of war to today. Simple as that.
Primarily weight and manufacturing simplicity. The fabric covered structures worked well for those applications and could be manufactured at a minimum of cost and complexity. It had nothing to do with protection against enemy fire - most all metal aircraft are thin skinned and vulnerable to even small arms fire.