I can only speak to experiences in Germany and Italy.

In Germany every aspect of life was affected by the war and it's fallout. Idk if you're American, but think about the American youth experience during the Vietnam War. You've got boys being drafted and sent to war from all over the country and they're experiencing that aspect of the war, right? And on the homefront you've got young people protesting and their brothers and friends are being drafted and some are coming back in boxes or coming back with PTSD. Music, TV, movies, and more. The war was being depicted or lived everywhere. It's recognized as a generation defining event.

Now think about Germany in WWI. An army of conscripts -millions of boys and men by the end of the war- and they're living through all sorts of horror and squalor. On the home front lots of people are starving because of the blockade and harsh winters, and a lot of the food that does get made is being shipped to the soldiers on the front! In addition you've got civilians working in war industries that are firing on all cylinders, all sorts of fundraising campaigns for the war effort, and their brothers are coming home in boxes or with shell shock. And then once they surrendered, all of those boxes, all the hungry nights, all the hours in the factory, all the marching, and all the money that had been poured in by every German suddenly became pointless.

The effect that this had on German society was clear, not only in diaries and letters from that time, but also in art and popular media like film. Weimar Germany had lots of great films coming out of it in the genre of "shell shock cinema." Because of a taboo of speaking about the wartime experiences in the immediate aftermath, shell shock cinema often used more abstract symbolism to talk about the war. Movies like The Cabinet of Dr. Caligari, Metropolis, M, and some might even argue Nosferatu. All of these films were influenced by the German experience during WWI. The Cabinet of Dr. Caligari is probably one of the more heavy handed of the movies, and it's free on YouTube. Aside from very forward thinking camera shots, use of shadow, and colored film, this movie is interesting because it came out in 1920 and dealt specifically with the way in which the German people, represented by the protagonist who ends the movie in an insane asylum, saw themselves in relation to the government/authority, represented by Dr. Caligari who runs that insane asylum. Metropolis deals with the evils of industry and capitalism, which was seen by some veterans like Peter Hammerer, as the force responsible for the war. In German expressionist paintings as well, the experiences of Germans were reflected by artists like Otto Dix who painted this harrowing image of mutilated WWI veterans. The effects of the war were visible and widespread.

In Germany a man's service in WWI was absolutely important. The experience of being a soldier in the biggest and most grotesque war the world had seen separated society into those who understood what it was like and those who didn't. Every German had a personal connection to the war, but the experiences of frontline soldiers were so extreme yet so commonplace that it defined a generation of boys and young men. The fact that it became so common in literature, film, and paintings of that time supports this. Were it not so, then

In the case of Italy, I actually have more experience reading from sources after the war, but these are still quite revealing. During WWI about 5.5 million Italian boys and men served in the military. 8,000 of them were volunteers. The difference here is that Italy was on the winning side. The arditi, who many people have come to know from Battlefield 1, were very proud of their service as an elite force that prevailed, unlike the German stormtroopers who, despite being just as brave, ultimately lost. It was members of the arditi, along with members of the mobile mountain troops of the bersaglieri, who followed Italian nationalist, author, and war-hero Gabriele D'annunzio into the city of Fiume to take it for Italy in 1919. These were well known and well respected parts of the military with which all Italians would be familiar due to wartime propaganda. Their experiences during the war have rise to the idea that Italy was owed the city of Fiume because of their sacrifice. Were it not for their wartime service specifically, they likely would not have done it, and likely would not have been greeted so warmly by the ethnic Italians who made up the majority of Fiume's residency. It was to these veterans that the early members of the fascist squadristi and blackshirts would look up. The demographics of the squadristi rank and file were quite young. R. Suzzi Valli found that, "nearly 90 per cent of the squadristi in Bologna and 83.5 per cent of those in Florence were aged between sixteen and twenty-seven, and that while the great majority of the squad leaders were demobilised junior officers, more than half of their rank-and-file followers were not old enough to have fought in the war." They had missed out on the event that defined their generation. For many of them, this was their chance to participate in the same struggle that their older brothers had: bringing glory to Italy. Giulio Teoni, who joined the blackshirts as a 16 year old in 1920, wrote "It seemed our turn had now come to continue the work of our elder brothers." So even if it wasn't important to the veterans themselves, it was definitely important to their younger siblings.

In the case of Italy I've generally spent more time looking at the first hand accounts of early fascists like Teoni, and very little time looking at those of veterans, so I can't speak to what the veterans thought of their place in society or the war's place in their lives. But by looking at the testimony of Teoni and others like him, as well as the data provided by Valli, i think it can safely be said that in the immediate aftermath of the war Italian civilians, particularly young boys, thought that the status of the veterans was important.

What i think would be an interesting follow-up question to ask would be: how much of a role did the status of "Great War veteran" play in Germany and Italy before the rise of the Nazis and Fascists respectively, vs. after?

I know this isn't quite what you were looking for, but it's as close as I can get you! Hopefully this'll at least give you an idea.

People i didn't expressly mention as sources:

Christopher Duggan, Fascist Voices

William W. Hagen, German History in Modern Times

All the stuff I linked to.