Interesting question. I've wondered the same thing. After the hits Puzo writes "The bloody victory of the Corleone Family was not complete until a year of delicate political maneuvering established Michael Corleone as the most powerful Family chief in the United States."

So it must have been important. Let's go back to the beginning. Why did Sollozzo/Barzini/Tattaglia need Corleone's political connections? Because without some sort of certainty regarding arrests, prosecutions, and convictions they couldn't profitably and safely run their drug businesses. Corleone was evidently the only boss who could guarantee that (as much as anything could be guaranteed in that business).

They basically extort Corleone to provide this protection and later pay him to do so. This gives the Corleones the chance to get even stronger politically/financially in NY, the East Coast and ultimately the US.

So all the political powers that be are dealing primarily if not exclusively with the Corleones, who are giving them even more money than before. It's a small thing perhaps for Vito to introduce Michael as his heir, and explain obliquely that there will be some minor changes upcoming soon but "that this is for us to worry about and has nothing to do with our friendship."
Perhaps after the hits go down police slow walk the investigation or start investigating everyone but the Corleones. Newspapers follow suit.

Also to bring Michael back cleared of the "false charges" and get ownership of the Nevada hotels and casinos despite Michael's clear criminal associations required some deft dealing. The Nevada businesses give the Corleones a legitimate facade and the capital to get things done on a national level. It becomes clear to most would be rivals that they can't run their business without Corleone say-so. Unrealistic? Sure. But that's what I think Puzo was trying to get across.