Michael's overarching goal was to find out who was the traitor in his family. Since he knew Roth was behind the Tahoe shooting, he needed to do just what he told Pentangeli: "Keep your friends close but your enemies closer." That's the main (but not the only) reason he went to Havana, with just a single (probably unarmed) bodyguard. Sending Frankie to meet with the Rosato brothers was something he had to do to satisfy himself of Frankie's loyalty-- like asking Carlo to admit he set up Sonny and to make him confirm that Barzini was behind it--even though he already knew the answers
Whilst I acknowledge Michael and Roth were playing their cat and mouse games, among others -
- neither Michael nor Roth thought the other will get out of Havana alive
- Roth gave his blessings to Michael pushing Klingman out because “Oh, Michael? Won't see him no more” [Tahoe shooting]
I believe, Michael showed his hand [without cause!] earlier on - Red flag that would have alerted Roth [if Roth was fooled prior] when Michael didn't have Pentangeli killed as he stated to Roth in Miami “Then Pentangeli is a dead man, you don't object”
Besides “sending Frankie to meet with the Rosato brothers” didn't advance his quest any further in finding out the traitor in the family
It ended up giving more ammunition to Roth eg: Senate hearing whereas if Michael had waited until his return from Havana – there was enough time! - then Michael's New York operation remains untouched and no Senate hearing either
Surely Roth is not going to tell Michael their Havana deal is off because Pentangeli is still alive?
Turnbull is a good man! My take "Frankie's loyalty" was beyond reproach
