Yep — not to mention, being, for example, Italian-American is a unique and special thing, a particular branch on the Italian diasporic tree as well as an important part of the US cultural landscape and history, a specific culture or set of cultures that split off from the various regional Italian-Italian (for want of a better descriptor) cultures that its originators came from over a period of time (with occasional new infusions), with those features recombining somewhere else in different ways, resulting in the culture developing somewhat independently — inevitably retaining some features that weren't retained back in Italy, developing new features, being influenced by its new surroundings, until it's something diverse within itself, and while identifiably Italian in flavour, nonetheless distinct and something to be valued for its unique Italian-Americanness. As I'm sure most Italian-Americans feel, and intend to communicate when using the word "Italian" to describe themselves.
If you've got significant Italian ancestry, an Italian surname, have Italian family recipes, cultural practices, religious practices, terminology, or whatever else in your life that can be traced back to Italian cultural origins, then of course it can make perfect sense to think of a component of your identity as being Italian in a genuinely meaningful sense, and for there to be contexts where saying "I'm Italian" would be correctly interpreted as "I have Italian heritage which I identify with in at least some significant way".
But it makes sense that when you're in a context where "I'm Italian" would usually be taken to mean something like "I'm from Italy, I speak Italian, and my culture is at least partly that of [a region of, or a group within] present-day Italy" — like, on an international webforum discussing European politics, or something — it would be clearer to describe yourself as Italian-American, because that is a different thing to what most of the world would understand by the phrase "I'm Italian". Not inferior, just different.
People don't get mildly irritated by it because they think they're better than Americans because they're properly German/Irish/whatever, they get irked because the shorthand that works well between Americans for discussing heritage doesn't translate well for talking to anyone else, so it can look like the American person is trying to claim an identity that doesn't belong to them.
It's not even that big a deal, that a few Americans occasionally discuss their heritage in ways that come across as a bit amusing or rude or unusual or whatever, to people living in some other countries. It's just a funny quirk that people have noticed some Americans occasionally have. People from other countries, including the UK, Ireland, Italy, etc., also often have particular quirks that others will notice are more common among people from there, and that will get commented on and maybe laughed at. I suppose it can feel more socially acceptable, or at least less risky, to comment and joke about quirks that Americans (and Brits, for that matter) have than about those of other nationalities, so maybe it could feel like Americans are uniquely targeted for their little foibles, but when I see joking about the way American individuals discuss their heritage, it doesn't usually look to me like vicious targeted sneering from people who hate Americans, just commenting on a minor thing people have noticed some American people sometimes do that seems funny or a bit weird.