Coppolla had to reduce the Godfather story significantly. Especially the parts with Johnny Fontane, Nino and Lucy.
Why did he keep the Woltz story? Just because the horse in the bed was so spectacular?
The Paramount producers were trying to push him to make a cheap gangster movie just to milk the success of Puzo's novel. FFC made it half an hour longer than he was allowed to.
My point is: The Woltz story is great but it has little or no impact on the general story.
We need the wedding scene to understand the family. The the story starts to unfold when Sollozzo meets the Corleones.
If we need the Woltz story then it is 1) to show how the Corleones work at that time and 2) how powerful they are. However, we know that already from the wedding scene: 1) The Luca/Johnny story, 2) the Corleones can buy a new law from the congress.