Sure thing Turnbull whilst “Vito's death was convenient to the script” Barzini can't afford “to wait until Vito died [not knowing when?!] to move against Michael” giving precious time to the Corleones to rebuild eg: Rocco's secret regime and get ready for battle
Gotta get Michael or Vito? That's the key for Barzini but Barzini was "chipping away at Corleone territories" small potatoes

It was pretty much common knowledge that -

Fish tank scene [private]
Quote
Vito: Well, Michael is now head of the Family

Moe Greene meeting [public]
Quote
Tom: the Don is semi-retired and Mike is in charge of the Family business now

The longer Barzini waited as Tattaglia said at the Commission meeting “as time goes by and his [Vito's] position becomes stronger....

Barzini and Tessio were slippin' and certainly underestimated Michael indeed - “weak” civilian Michael, killed Sollozzo and McCluskey single handed when “nobody has ever gunned down a New York police captain Never”!

As you always say [or similar] Trilogy theme – underestimating / overestimating Their Achilles heel