Interesting you say that. Having spoken with a number of people who lived under Soviet rule, many will say that compared to Soviet methods, American methods of propaganda are more sophisticated. The average American may not notice as easily because like the proverbial fish he does not know what water is.
I don't deny the Americans are good at their influence operations too and maybe more subtle in things like saying they are pushing for freedom and democracy while not always exactly doing that. The Russians seem to go more for claiming black is white and Ukraine started the war rather than they did and such like.
I'm honestly curious what those methods may be and what the the target audience for this propaganda is? You're mentioning that average Americans would not notice, so is this American propaganda targeted at Americans?
https://news.ycombinator.com/item?id=43736805 is one example. Another is the Highway of Death, which inspired a Call of Duty level, but swapped the perpetrators from US to Russians [1]. Yet another might be A Time to Kill [2], based on a novel [3], inspired by a true event. Again, perpetrator and victim were swapped - in the novel and film, there are two white attackers, and the victim is black. In the real event that inspired it, the attacker was black, and the two victims were white. The subway killing of the three scoundrels in the Joker film was likewise inspired by a real event [4], but again, the antagonists were changed from black to white. Or how Dr. Who deliberately alters historical settings [5]. Especially amusing is the case of The Sum of All Fears - in the film, the villains trying to construct a nuclear bomb are neo-nazis. In the book it is based on, the villains were Palestinian terrorists. Finally, in the real event the book was inspired by [6], the uranium was stolen by Israel.
A more straightforward example is the many times that the New York Times 'mistakenly' claimed Iraq had expelled nuclear weapons inspectors, in the run up to the Iraq invasion [7]. This is more obviously false/a lie, while the prior examples use the cover of fiction to mislead.
'Small' changes, but pervasive, like water, mostly cut only one way, and they add up.
[5] “And we’ve kind of got to tell a lie: we’ll go back into history and there will be black people where, historically, there wouldn’t have been, and we won’t dwell on that. We’ll say, ‘To hell with it, this is the imaginary, better version of the world. By believing in it, we’ll summon it forth.’ - https://www.doctorwhotv.co.uk/moffat-on-diversity-in-doctor-...
There is also government propaganda, yes [1,2] - movies that want to use army assets need script approval from the military - but I don't know, and don't think, this was the reason for most of the distortions I listed (nor would I say any except the Highway of Death one were nationalistic). In my opinion, the simple shared ideals of most in the entertainment business, without collusion or coercion (if we discount the leverage of finance from coercion), make for the far more effective propaganda. And occasionally, the propaganda is clumsy enough to get caught - such as when Roald Dahl's books were altered [3]. The list of changes [4] makes for interesting reading.