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.

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Yes, but not just because it was graphihc and spectacular. See below:

The Woltz episode has no further impact on the development of the film. But: 1) and 2) are just stories, with little dramatic impact. The Woltz scene shows the crude power that backs up Vito's maneuvering and diplomacy. It shows that Vito can reach across the country and use violence to get his way, even if it's only to satisfy his ne'er-do-well godson. But there's also a business interest: Vito controls or influences Hollywood labor unions and needs to assert his power over Woltz to maintain his credibility 3,000 miles from his home.

It was also meant to illustrate Vito's intelligence. He analyzed Woltz's strenghts and weaknesses and decided how Woltz was most vulnerable.

It also provided great insight into Vito's (via Tom) way of doing business. He tried to make a deal, offering help with the actor's drug problem and labor issues, but Woltz was too bull-headed to see it. He was intent only on thwarting Fontaine, no matter what sort of deal Tom was able to offer on Vito's behalf.

Woltz was loud and insulting. However, Tom maintained his calm and seemed completely beaten and accepting of Woltz's decision. And then he got his revenge.

Interesting detail: During the dinner scene, Tom hardly says anything. He mentions the sacred meaning of being a godfather. And that Vito never asks twice for a favor. Then Woltz starts in fact a monologue. And Tom doesn't interrupt at all. He waits and leaves.

That scene is absolutely indicative of the training Tom received from Vito. Never get angry; never threaten. Always use reason. Tom remains calm also because it doesn't matter what Woltz says; Vito promised Johnny the movie role and Tom knows that he's going to get it one way or the other. So, Tom lets Woltz vent while completely confident that nothing will come of it.

His wine glass is filled two times in a row by the black maid as well.

Thinking about it now, i bet everyone hears knows that.

It shows the whole concept of the fist hidden by a velvet glove. When Woltz insults Tom at the studio, Tom simply says he admires his films. When Woltz shows Tom the horse (and the key to the success of Tom's mission) Tom simply admires the horse and drops the subject. After Woltz' dinner rant, Tom simply thanks him and asks for a ride to the airport. This is all vintage Vito, as is the "head in the bed" scene. This is in stark contrast to Santino's leadership style, and this scene sets this difference up.

Yes. Puzo makes the point, more than once, that Vito taught Tom never to make a threat. Early on, during Vito's rise, he gets a contract to truck booze from a bigger criminal enterprise. They accompany the contract with a threat, which "diminished" them in Vito's eyes. The best example was when Tom called Carlo after Sonny's assassination. He said, in effect, "Some men will come to your home. Just tell them to call me. Don't say anything else. No one will blame you for Sonny's death." Puzo wrote, "The Don had always taught Hagen never to utter a threat, but Carlo got the message: He was a hair away from death." Nicely done!

The horse's head had been the single-most discussed section of the novel.

And it may seem curious now, but a major factor of the novel's commercial success was its gossip about Frank Sinatra -- specifically the Tommy Dorsey and Maggio legends. This may have had as much impact on the book's bestseller status as the inside info about the mafia.

One of Coppola's most significant decisions in the entire production process was dumping virtually everything about Johnny Fontaine except the Woltz story.

So people got to see the most sensational aspect of the novel visualized, but everything else about Fontaine -- as well as Lucy Mancini/Dr. Segal (an especially superfluous story arc) -- was wisely dropped.

Puzo spent a lot of time in Hollywood, and he used the Fontaine and Nino stuff to show off. Wisely dropped is right. That Hollywood garbage ruined "The Last Don." One of the main reasons the TV film was better than the novel is that the Hollywood stuff was mostly edited out.

Woltz' rant reminds me of Mitt Romney.

The line "Johnny FOntaine doesn't get that picture," was lifted from a conversation between FFC and the head of Paramount who did not want to use Al Pacino as Michael.
The studio wanted Robert Redford, which IMHO would have been a terrible choice....not that Redford was bad, but a blonde Sicilian?

Redford would have charmed Sollozzo to death.

Isn't it when am i going to California?

In a deleted scene, after Tom asks "When am I going to California?", Vito answers him:

Vito: Tom, you're not Consigliere yet. And to tell you the truth, I don't know if you will ever be. But I do know one thing for sure. When I tell you to do something, do it, no questions asked. 25 years or not in my house; Sonny's friend or not. I'll cut out your stinkin' Irish heart in a minute, throw it in the fire, and dance a jig around it. Capite?

Tom: I ... well, I was just asking because I have to try and get a priority pass from that Pentagon officer friend of mine to fly out there and it's the weekend and all.

Vito: Oh. I see. Well then, my mistake. Too much wine. Love ya Tom.

I have never considered Tom's offer to help Woltz with his drug problems and labor issues as if they currently existed. It was more of a threat, that if Johnny didn't get the role, he would have those problem arise and destroy his company.