My take:

Sometime after GF III, possibly in the early 1960s, Altobello is named by Michael as the new Don in NY, the replacement for Pentangelli as it were. He is an older man, and seemingly, more savvy to "big deals" than Frankie was (which helps Michael in his quest for legitimacy). He is given the position with the promise he will have more leeway to run things his way day to day than Frankie had, and eventually, comes to be seen as a Don in his own right as such after almost 20 years in the role. He was close enough to Vito to be named Godfather of his only daughter.

I do not think Vito would've made a soldier from a rival Family his daughter's godfather given the importance of such a role. No, only a trusted figure in his Family, I would think.

Let us say he is a young man when Connie is born, say in his 20s, perhaps a trusted but young button in Vito's Family, that would make him be around his 50s or 60s in the 1960s, closer to Michael's generation than the Mustache Pete that Frankie was.

Only, that's the inherent problem. A man like Frankie saw the small picture and as such was loyal. He was literally a man who was best at taking orders, doing numbers, drugs, gambling, and such; not the grand designs Michael had. Altobello on the other hand; He is of Michael's cloth and while he plays the role of a good ally, perhaps over time he secretly cultivates to be what Michael is, or surpass him. Where Michael sees big deals, Altobello sees even bigger ones, and comes to see Michael over time as an obstacle to them.

During the 1960s, he takes Zasa under his wing and Zasa becomes HIS de jure Boss of the Family (But in reality, as we see in the Commission scene, Joey is indeed just a pawn in the larger game to be rid of Michael), acting as if he himself has approached retirement, an elder statesmen of the Mob, but in reality, Zasa is his front Boss - his bella figura, as he himself puts it - a clownish figure intended to cover for Altobello, his lighting rod to take the hits publicly. It is clear no one takes Zasa seriously in the Mafia world; and he is not lamented or missed by Altobello when he is taken out. He played his part and took the bullets that should've been pumped into Altobello's belly, as was probably the design all along.

Altobello having been made Michael's Street Boss would explain his seeming deference to Michael, as well as the fact that he "backs" Zasa and seems on the surface to have control over him, as if Zasa is his apprentice. It'd explain why Altobello is on the Commission, and Zasa is not.