I'm not sure the word 'allowed' is appropriate...but anyway at our first introduction to Fredo (at least in the film) he is visibly tipsy when saying hello to Michael and meeting Kay. So one would understand why there would be no room for him in the Don's office during requests on his daughter's wedding day.

Second...Fredo was present at the initial meeting between Sollozzo & Vito and say what you will about his intelligence, but he at least knew not to speak out which is more than can be said for his brother Santino.

All in all, I agree with SC that up until the time he was shot, Vito might have tried in some way to give Fredo some kind of exposure/role in the Family Business. Eventually it became clear to all that he was incapable of handling any of it.

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