Wasn't it the Irish that developed rounders? I'd agree that there are obvious similarities, but I'd say Baseball has truly developed as it's own sport. I'll say the same for Rugby and North American Football as well. But I know what you mean. Even Ice Hockey has it's obvious roots with European Field Hockey.
Well, technically, rugby is the older brother, if you want to look at it that way. But American and Canadian gridiron are most close to each other, so I would consider them to be brothers, and rugby to be their parent. Gridiron developed out of rugby (so in that sense rugby is the parent), but this happened at an early point in rugby's evolution. It's one reason why we call touchdowns, "touch downs" even though gridiron no longer requires players to touch the ball down. It's a holdover term from rugby. Canadian gridiron held on to a lot of the rugby aspects of the game longer (they even called their governing bodies rugby unions for a long time), but they always ended up adopting American gridiron ideas (like the forward pass), sooner or later, so that the two games are now very similar.
Rounders was played in both Britain and Ireland. It was one of those amorphous ancestral games without any obvious origin (it was also sometimes called baseball; the names were interchangeable). There are people who theorize that the Irish immigrants to the USA in the 19th century preferred rounders to cricket, and that they helped switch Americans from playing cricket to playing the various ancestral forms of baseball/rounders/townball, etc. Maybe. When the Gaelic Athletic Association was formed, though, one of the first sports they sponsored was rounders, along with hurling and gaelic football. GAA rounders apparently has different rules from the version of rounders played in England. But they are both pretty similar to each other, so I understand, and very different from modern baseball.
Seriously. Callign our teams "clubs" doesn't strike me as Eurosnobery at all. It doesn't even strike me as European. It's perfectly normal in America to call our sports teams -- of any sport -- "clubs". I know I do it all the time referring to, say, baseball teams.