When last we saw Hagen in GFII, he was at odds with Michael over Michael's plan to eliminate Pentangeli and Roth. But in III, Michael's the one who's trying to avoid violence. Perhaps if Hagen were in III, he'd be vindicated. Then we could have had a nice subplot with Hagen counseling Michael against bringing Vincent-- with his hot temper and his penchant for violence--into the Family.
Also at the end of II, we saw how Neri was trying to deposition Hagen to be Michael's #2 guy. If Hagen and Neri both survived into III, we could have had a seriously Machiavellian contest between the two of them.