Dual citizenship depends on the country, but for example with Ireland, people can get an Irish passport based on Irish ancestry but they still aren't allowed to vote there. And I think that's an important element because it's indicative of the major problem - once you leave the country of your origin you no longer have a stake in its future. I don't have to pay the taxes or follow the laws they pass. My stake is in my community here in America.
Now as a first generation person I still have enough family there to at least still care a lot, and even a 2nd generation person might still say that, but at some point even that link is broken and it certainly is for most Americans/Canadians whose families came over 100+ years ago.
And what happens is that culturally the 'old country' moves in a different direction than the people who migrated and their descendants who start to adopt the ways of the places they moved to and also 'miss out' on the major changes that happened since they left. Even as someone who spent a huge chunk of my life in the 'old country' and hasn't been gone for a hugely long time (and may yet go back at some point), I notice a degree of estrangement from the place every time I go back. Remember why they say that you can never go home. There's a truth to it.
Right, it depends on person to person but there's more to being from somewhere than 'Can I go to Grandmas if I land there?' albeit I agree that time affects the relationship, just not to the extent that you're suggesting. Kinship is a real thing and as I said earlier, culture lives on in foreign countries as well.
I don't buy the voting argument though. I can vote in France just because I got a passport through my grand-mother. That doesn't make me more French insofar as I still don't really feel French because I had little exposure to it growing up besides TV and books. But tell my mom (who has never lived in France either) that she's not French and she'd rightfully flip because it's her mom we'd be talking about and she (rightfully) wouldn't give a hoot what some random Frenchman would think. My dad is a born and raised Moroccan and tried to live there for a couple years maybe a decade ago after spending 40+ years in Canada and the US and struggled to adapt. His Arabic is still as impeccable as his French. Does that mean he's no longer Moroccan even if he still has Moroccan friends, siblings, follows the news, eats the cuisine, etc? I think that'd be an absurd suggestion.
Not everything is just one thing and millions around the world embrace all aspects of their lived experience.