From Mein Kampf it seems like he considers it a fairly minor accomplishment - he served because he was emotionally compelled to, and he wasn't superior to anyone because he risked his life for Germany or suffered a gas attack and lost eyesight for ~2 weeks. Not that he was especially brave or distinguished in his short career, but in the US today even the lowliest private is respected for their service. Did he mention this a lot as the leader of the Nazis? Was it common knowledge and a political advantage for him?
Well, American soldiers get put on a pedestal by a lot of Americans, much more than other nations do their soldiers. So to compare the two is difficult. What makes it even more difficult is that the German society after WWI was very different from the German society before the war.
Every aspect of German life had been directly affected by the war. Men had gone to fight, and many hadn't come back. Some were closer to boys than men. Women had worked in factories, and families starved because of Britain's blockade. The entire nation was traumatized by the war because all of the lives lost, the insane amount of money spent, and the immeasurable effort of the whole population had been for naught. So in post-war Germany, conversations about the war were difficult, and understandably so considering all of the suffering.
Being a German veteran of WWI wasn't really special, because plenty of people were, so what? Everyone was facing hardships during that war and oftentimes, civilians felt like the soldiers had it better in a way, because the soldiers had food, while the soldiers thought the civilians had it better because they weren't on the front lines. This is particularly well depicted in the German WWI film Westfront 1918, which came out in 1930.
But what about Hitler specifically? He joined up out of what I'd call ethno-nationalism, specifically pan-Germanism. He was from the Austro-Hungarian Empire, but at the time Austrians were considered ethnic Germans (today there's more controversy). He had actually dodged the draft in Austria-Hungary, partly because he didn't like the influences of the Hungarians and Slavs within the empire. He served as a message runner which was, as we know, incredibly dangerous, he got gassed and suffered "hysterical blindness" for about two weeks, and he received both an Iron Cross 1st class and an Iron Cross second class. Earning an Iron Cross 1st class was honestly a decent accomplishment.
Was that a big deal to Germans by 1933? Not particularly, no. WWI veterans were all over the place, so he wasn't special in that regard. What overshadowed his service even more, was that the president of Germany who assigned Hitler as chancellor was literally former General Paul von Hindenburg. If you want to stand out as a leader with military service, it'd be pretty hard to follow his act. He and Ludendorff were THE guys who ran Germany's military by 1918. Hitler's Iron Crosses couldn't show that up.
Sources:
William Hagen, German History in Modern Times: Four Lives of the Nation
Adolf Hitler, Mein Kampf