Feels like there's different reasons logistically based on the sports, but the main answer feels like it's as simple as it's never been like that just cause that's how they were initially set up.
But, I think a big difference is that other leagues in the U.S. are used as developmental leagues a lot. There's not enough of a talent pool in american football given the much larger roster size vs other sports to have much more than the 32 teams in the NFL imo. Any other pro league has a pretty low quality of play/competition from what I've seen. Baseball and hockey both have extremely vast networks of developmental teams, a lot of which are directly affiliated with the main pro team. For instance, the NHL is the main league in hockey, the AHL is the minor league where teams have a direct affiliation with an NHL team and are used in large part to develop players for the main team. Then there's several junior leagues in Canada (OHL, QMJHL, WHL, ect) and the NCAA in the U.S. which are exclusively younger players that aren't directly tied to a team, but are often metaphorical breeding grounds for new pros.
I'm not familiar with how most sports leagues in other parts of the world (apart from hockey) work, but I'd imagine that the process of a recruit/prospect isn't quite the same as most American sports leagues.