Is there any reason that we (the audience of the movie, solely) should know that Genco was consigliere besides the reference Santino makes to him while arguing with Tom?

Not really.

There is a major deleted scene that has Vito, his sons and Johnny visitng Genco on his deathbed immediately after Connie's wedding.

In the special Television version of The Godfather Saga, after Connie's wedding, we see the Don,his sons and Johnny Fontaine leaving to visit Genco. At the hospital we see Vito ask Michael if Kay was able to get home all right, and then the Don looks at Michael's military decorations and says something about the miracles that Michael has performed for strangers. Then he tells Michael that he has plans for him after his graduation. Then in the hospital room we are shown a dying Genco who begs Vito to stay with him because he believes that Vito will somehow stop death from coming.

No, there is nothing the viewer gains from knowing that Genco was the prior Corleone Consigliere other than, as you point out, possibly understanding the reference to him voiced by Sonny to Tom.

But we see other references: The sign of the Genco olive oil company when Sollozzo meets Vito and a sign "Genco real estate" when the Corleones leave NY.

Well when you watch GFII, you are shown a young Vito working in a grocery store and eventually losing that job due to Fanucci shaking down the owner of that store by forcing him to give his nephew a job. This causes the owner of the store to have to let Vito go because he cannot afford to keep two employees on staff and at the same time he cannot say no to Fannucci or he will be out of business.

It's quite obvious that Vito feels indebted to the owner of the grocery store and his family for basically taking him in and providing him with a job for as long as he did.

So later on when Vito does eventually rise to power he basically repays the owner of the store's son by 'going into business with him' ;)....Genco Olive Oil Company....and making him his second in command (consigliere) in the family business.

So actually there does wind up being some significance in us learning that Genco Abbandando was indeed Don Vito's consigliere.

And with GFI as a stand alone, it's purpose is to show us that the Don's Sicilian Consigliere was on his death bed and would eventually die and was being replaced with the Don's foster son, who happened to be a NON Sicilian and instead of German Irish background. That probably was the real significance in FFC's reason for briefly mentioning and introducing Genco..... It opened the door to set up the sub plot in the character of Tom hagen, of a Non Sicilian, becoming Consigliere.

With respect to GF alone: Being introduced to the name Genco and the reference to being consigliere, provides the audience with background information and it serves to enrich the character relationships and plot. As DC points out, the Genco reference is clarified and expanded in GF 2.