I would agree that the term “Latino” is dumb and really shouldn’t be used as it doesn’t account for, say, indigenous people in Latin America who don’t speak Spanish. Generally speaking, though, it’s a term to to broadly coat a people born in a specific region influenced by a particular culture (ie. Spain and Portugal). Besides, from my experience most Spanish speakers who are recent immigrants don’t talk about being “Latino” at all and just refer to their mother country instead (“I’m Mexican/Puerto Rican/Peruvian”).
To your second point, I think people get hung over with the language because even though we aren’t from Spain our culture is heavily influenced by it. Mexico, for example, is quite literally a blend of both Spanish and indigenous customs; many of us are also racially mixed between the two (as we have the historical term “Mestizo” to indicate Spanish and indigenous mixing). Unlike the US, miscegenation was widely more practiced in Latin America, and since Spain has been heavily influential in the Americas it’s only natural that we adopted their language.
Which brings me to your last point. When I was a child, I remember being teased a lot by my family for getting words wrong or not knowing how to speak fluent Spanish. I didn’t feel bad or angry about it but it was something that made me realize that I was different from them. To them, I was the Americanized boy who, like others on this forum, liked video games, American movies and music, and had friends from all over the world who all spoke English. When I got older, before I graduated college, I worked with my father at a large grocery that employed undocumented immigrants mainly from Mexico. The same thing happened there. I was teased a lot for being “too American” and for my inadequate Spanish (I could speak it but I wasn’t confident with it as I had to at times insert English for words I couldn’t convey). Through time, however, as the years went on the more I worked closely with these guys the more I started to notice my Spanish improving and I started to speak like them; I started using heavy Mexican street slang and I got so good and natural at it that eventually the teasing stopped and they accepted me as “one of them”. And this has translated to my family. Improving my Spanish made me feel closer to my families culture. When you learn to speak a language you also in a way learn how to think like them. So while it sucks that there are situations where, for instance, American born Latinos are criticized for not speaking Spanish, it does beg to observe that language matters, and speaking a language helps in not only communication, but building relationships, camaraderie, community, and a sense of belonging. Per your example, how could a person ever build a relationship with their grandparents if they can’t even communicate with them verbally? I personally feel that learning a language is the ultimate sign of respect and best way to preserve a culture.