Were the helmets of the time given to soldiers to make them feel less vulnerable? Enemies can still fire at the much larger and easier to hit body. Is there any research that showed that helmets saved lives on an effective scale?
First, soldiers hide. When shot at, or under artillery fire, unless there is a need to keep moving (and sometimes even then), soldiers will hide in the best available cover, or as time permits, make their own cover by digging in. However, the head is often poorly protected by cover (including trenches, foxholes, etc.), because it is still important to observe the surroundings and to fire at the enemy - both of which often expose the head. The main items of protection for the soldier in the World Wars were the helmet (for the head) and the shovel (for the rest).
Second, the big killer in both WW1 and WW2 was artillery. For WW1, artillery caused somewhere between 70% and 95% of casualties (the variation depending on exactly what is counted as a wound, and how multiply-wounded soldiers are counter). Stopping fragments and shrapnel from artillery fire was the main job of the helmet. A properly bullet-proof helmet would have been nice, except that it would have been too heavy. Helmet were not intended to stop rifle bullets (but could stop pistol bullets).
So, how well did they work? This has been studied (variation in collection of statistics (e.g., it wasn't always reported whether wounded soldiers had been wearing helmets) means that the results aren't quantitatively exact, but they still give a good indication). One nice compact study by the US Army Operations Research Office (Hitchman, N. A.: Keep Your Head . . . Keep Your Helmet. Army 8:42-44, September 1957) is summarised in James Boyd Coates, Jr. (ed), Wound Ballistics, Medical Department, United States Army, 1962, online at https://history.amedd.army.mil/booksdocs/wwii/woundblstcs/DEFAULT.htm (see chapter 11, pp 647-648):
Of all hits upon the helmet, 54 percent were defeated.
For every 100 men wounded while wearing helmets, 9.6 men received wounds in the cranium. Without the helmet, it would be expected that 11.4 [additional] men would be wounded in the head.
The M1 helmet prevented a number of incapacitating hits equal to 10 percent of the total hits on the body.
The estimated savings in total battle casualties means that the helmet in World War II probably prevented wounds in more than 70,000 men. A significant proportion of these men would have been killed had the helmet not been worn.
To get the same amount of saving by protecting other regions, body armor weighing more than twice as much as the helmet would have to be provided.
Was this saving lives on an effective scale? Over 22 million M1 helmets made during WW2, and 70,000 casualties prevented. For soldiers hit, having the helmet stop over half of things that hit it is significant.