It’s a devastating coda that underlines the tragedy of this family in general and also looks at Michael’s constant struggles against the grain of the family.
I mean first we learn that it was Sonny who introduced Carlo to the family, and therefore to Connie, which gives greater depth to the events of Part I.
Then you sort of mentally tally up everyone in that scene who’s now dead: Vito (natural/unnatural causes), Sonny (killed), Fredo (killed — by Michael), Tessio (traitor, killed) and Carlo (traitor, killed). The only two survivors from that scene are now people Michael feels he can’t trust anymore: Hagen (pushed away by Michael’s paranoia) and Connie (who went behind his back letting Kay see the kids). Mama Corleone is passed on. Apollonia, dead. Clemenza, dead. Pentangeli is dead. His unborn son, aborted. He still has his kids, but without Kay, there’s a distance there.
His mob family is crippled, his real family has almost disappeared.
The final shot of Michael alone is just that. He’s Alone with a capital A.
And then what the viewer has to do is go back to him talking to Mama Corleone:
“What did Papa think — deep in his heart? He was being strong — strong for his family. But by being strong for his family — could he — lose it?”
And then you look again, young Michael, alone at the kitchen table and then older Michael, alone at his estate, and you think, yes, you can lose your family.