Maybe I've got a criminal enterprise nature, but I there wasn't anything I didn't understand the first time I saw the film. Of course, I had already read the novel.
Alot of the lack of understanding you express can be a function of one's age and experience watching movies and with life. For example, if you're watching Gone With the Wind, a four hour + movie, you certainly wouldn't expect that from beginning to end that the Civil War was fought and concluded in just over 4 hours. That is a function of your knowledge of history.
Also, Don Barzini is greeted by Vito at the wedding. Both Vito and Sollozzo identify the Tattaglias at their meeting. So, you know who they are. If the wedding is taking place outside in sunshine, one might very well conclude that it is taking place in the summer. If one then sees Christmas decorations in a following scene, one might very well conclude that it is now around Christmas and that several months have passed since the wedding. Tom tells us that it's almost 1946. Well, for those who know their history, they are aware that WWII had just ended a few months ago. So, they understand Michael's appearing in a uniform at the wedding. Also, why conclude that Sonny is living in an apartment and not a house?
A filmaker doesn't have the luxury of unlimited time. A four hour movie is pressing it. Those are just a few points to ponder.