Good question, Capri! And--miraculously--one that hasn't been asked before.
I think the answer is in this bit of dialog from the fishtank scene:
MICHAEL (seated on a chair)

After we make the move to Nevada -- you can break off from the Corleone Family and go

on your own. After we make the move to Nevada.

CLEMENZA

How long with that be?

MICHAEL

Six months.

TESSIO (to Don Vito)

Forgive me, Godfather, but with you gone -- me and Pete'll come under Barzini's thumb

sooner or later...

Sure. Vito, old, in declining health, didn't have long to live, Michael, in Nevada, would be preoccupied with building his "legitimate" businesses; and Tessio underestimated him, If Tess and Clem really did form their own "family" out of the Corleones, they'd either split. it in half, or fight each other for sole possession--either way weakening the entire enterprise. They'd be sitting ducks for Barzini.

I believe Barzini didn't want to move against Michael while Vito was alive because he still feared Vito's cunning and political influence--killing Michael would rally him and his still loyal legions. Barz might have won that war, but it was easier just to wait for Vito to die. With Vito gone, Barzini figured he had a green light to kill Michael. Why? Doing so, right after Vito's death, would deal a body blow to Tess and Clem, improving his negotiating position and inevitably causing one or both to come to terms with him, Barz may have been making plans to move in on Nevada gambling. The Corleones were the big power in gambling and already had a toehold in Vegas through their investment in Moe Greene's hotel. By eliminating Michael, he'd have a freer hand to move on Nevada gambling.