Sometimes works of art (paintings, music, film, sculpture, architecture, literature, doesn't matter) are so profoundly influential as to become a part of the fabric of that medium. I think the Godfather is one of those films that inspired an entire generation of filmmakers to weave the special bits into everything they created since.
The problem with watching it now is that the craft of filmmaking has spawned from it and molded around it, and the things that made it special are now mundane. Try to watch Citizen Kane, or 2001 A Space Odyssey, or Seven Samurai, and you'll see every trope and flaw because their impact is no longer unique. But that's not because they weren't amazing films, it's because they have all be copied and modernized and lampooned to death.
With the Godfather, a film buff could talk for hours about the lighting, the symbolism, the mise en scene, the music, and how it was all seminal to half the movies made since. Watching it with virgin eyes, though, and you'll see reflections of Goodfellas and Casino and Scarface and Once Upon a Time in America and The Irishman and A Bronx Tale and Donnie Brasco and New Jack City and Road to Perdition and We Own the Night and The Departed and The Untouchables and probably 50 other movies I can't think of off the top of my head.
You can't help but see it as a relic, a source of inspiration for the movies you saw before and loved. That's why you don't see it the way they did, and why it seems over hyped.