These sorts of questions are notoriously difficult to answer because of course there was an incredible diversity of opinion.

A few narrative points first.

The Russian Civil War following the October Revolution lasted until 1922, so given that your question asks about the 1920s it is hard to ignore the fact that there was a war on. There were plenty of "everymen" among the whites.

Second, you mentioned the "Soviet Union began to lean towards the Axis powers leading up to WWII" and I frankly don't know what you are referring to. The Soviet Union was never on good terms with the Axis powers. The treaty between Hitler and Stalin never indicated an alliance, it was an agreement not to attack one another - which both knew was essentially inevitable. Indeed, the lead up to World War II was accompanied by a lot of anti-German propaganda and swell of pride (see David Brandenberger, National Bolshevism for more on this during the 1930s).

But on to your main question:

There was certainly a lot of revolutionary enthusiasm during the 1920s. In hindsight it can seem almost naive, but there were quite a lot of people who really embraced the idea that they could help build a new world. That is a powerful idea. Richard Stite's Revolutionary Dreams discusses a huge number of utopian/revolutionary experimental social movements that embraced revolutionary ideology - although that meant different things to different people sometimes.

Stalin's so called Cultural Revolution of the late 1920s and early 1930s put a damper on a lot of these experiments and more clearly and rigidly defined acceptable social/cultural forms. Stalin's purges also made things more dangerous for people, as you might imagine. In some ways though after the Purges, the war helped keep the opinion of the state high. People genuinely rallied around Stalin and the Red Army during World War 2 against the Nazis.

I'd also be doing a disservice if I didn't mention non Russians. You asked about the "Russian/Soviet everyman" in the question, so I'm not sure if you meant you wanted an answer only about Russians or not, but it's impossible to talk about the Soviet Union without talking about non-Russians.

The Soviet state's relationship with the other SSRs and with national minorities is really complex and a many books have been written on the subject. The "national question" posed a difficulty for the Bolsheviks and they new it from the beginning. Imperial Russia had a "chauvinistic" relationship with them, and the Soviet leadership wanted to change that - at least on paper. There was a very odd mix of national autonomy and centralized authority that took shape. Again, the opinion of the Soviet state varied greatly by region. The Ukrainian relationship was tumultuous, made worse by the famine of 1932-33. The holodomor, as it is called, remains a controversial part of the history that I won't get into here, but insofar as the "everyman" was concerned in Ukraine at the time, it certainly hardened their attitude towards the Soviet leadership.

And I haven't even talked at all about Central Asia, or the Caucuses, or even broadly speaking the peasants in Russia for that matter yet.

The peasant issue is practically its own question - but if we are asking about "everymen" then the peasants surely need to be mentioned because they represented the majority of the population. Things like collectivization created an animosity between many peasant villages and the state.

You can see what it is so difficult to talk about an "everyman" in the Soviet Union when it was so large and so diverse. The experience of the early Soviet Union depended greatly on where you lived, what your class background was, and what your experience was BEFORE the revolution. There is no doubt that many people welcomed the revolution and the changes the Bolsheviks brought. It offered them a lot of opportunities they had been previously denied. Many people genuinely bought into the project well into the 1930s. On the other hand, there were many people for whom the October Revolution was not welcome, but the particular character of their feeling or opposition to the revolution was by no means uniform.