"Tessio under wraps" was a creature of the novel, not a point made in the film. Vito needed Tessio present to hear his unequivocal "no" same as the others,
I also think that, even in the novel, the mysterious Tessio was mostly a creature of the 1930s. Vito's main purpose in giving Tessio a free hand was to minimize contacts between Tessio and Clemenza. That purpose would remain even after a couple of wars had revealed that Vito and Tessio moved hand-in-glove.
