There are several problems there:

• Stereotypically, the Americans doing this are way further removed from their ancestry than the second-generation immigrants you describe (in fact it's completely normal and accepted for second-gen immigrants to identify as their parent's nationality as well in Europe);
• "I'm Italian" and "I have Italian ancestry" are NOT the same sentence. You seem to realize that, but many Americans don't, and the comment you replied to complained about the former, and the difference is fundamental;
• Europeans are generally not on board with the whole "ethnic identity" stuff that Americans do, for a variety of reasons that one could simplify down to "last time we did that, nazism happened". The mainstream progressive view is humanist and intentionally colorblind, and it is therefore profoundly shocking to see Americans derive a sense of self-worth from their blood, because these are the talking points we normally only hear in documentaries about Mussolini...
  Now I have spent enough time reading about how American view their complicated relationship to race, ethnicity, and ancestry, to understand where you're coming from, but this is fundamentally at odds to the humanist approach of "we're all the same and who your great-grandparents were does not define who you are in any way". (Which is obviously idealist, and does tend to "whitewash" some struggles, but it is nonetheless the prevailing approach).

Mate, I've seen long-term immigrants not just of my own nation but other nations who returned and was even myself an immigrant of my own nation for over 2 decades abroad, and after 2 or 3 decades people living abroad are already culturally and even in values different from their countrymen, due to a mix of partially absorbing the values and way of being in life and society of were they live, and because their own country kept on changing over time generally in a different way in which they themselves change (it's quite funny how they have ideas about how their own country of birth is that don't really match the reality and look silly and outdated to the people actually living there).

This is a mere 2 or 3 decades for people who actually grew up in their nation of origin.

People 2 or 3 generations away from said nation are not only descendants of immigrants with a deviating cultural framework as describe above, but they have grown up in a different nation (and from all my observations living in a couple of countries, people culturally tend to be closer to the country they grew up in more than the country of their parents) and at least their parents and possibly their grandparents were already people who grew up in a different nation and only knew about the nation of their ancestors via 2nd or 3rd hand accounts.

Whatever "culture" and "value" they have from their ancestors' nation of origin is a thin slice, deeply degraded (often charicaturally so - note the mention of spaghetti eating to mean "culturally italian", something which would make me Italian and my Italian ancestors if any came over during the Roman Empire) and severelly outdated (a century or more) version of the culture and values of the nation of origin of their ancestors.

The difference for example between an American of Italian ancestry and one of Irish ancestry is token if that much compared to the difference between an actual modern Italian and an Irish: American-Italian, American-Irish and so on are but sub-cultures of the United States of America culture and draw most of their ways and values from that one, not from the cultures of the countries of origin of their great-grandparents.

As long as you speak the language, it's fine by me. Once you stop speaking Italian at home (in this example) it's over, you can't call yourself Italian anymore.

According to the Codex Wiesonius.

funny that you say that, not all Europeans are stuck in the same nationality for 10 or 30 generations back, maybe not even majority.

My great-grandmother was German, never learned the language of what is now my nationality. My grandmother and her child (my parent) didn't speak German and have never subscribed to German nationality, neither do I (but I speak a little bit German though becouse of school not because of family). Maybe it's because the identity of the place I live in is as strong as Germany's so it's a simple choice. But for a country, whose entire schtick is "'Murica fokk yea" I am sometimes baffled how much this ancestral identity matters among people who are supposed to benefit from the whole thing (white middle/upper classes).

No, this is an American thing. Other countries in the American continent have the same immigrant thing going on and we don't call ourselves Italian or whatever. We are all from the country where we were born.

Culture persists in some way for generations, you know. An immigrant's grandchildren probably won't fit in in their home country, but they're still distinct from mainstream American culture.