It wasn't personal. Simply business.

Alot of people didn't have any faith in or respect for Michael due to his inexperience. The Barzini/Tattaglia alliance had already opened up two "book" operations in Brooklyn the day of the don's death. In their eyes, Michael had not earned the title and his succession was questioned. For Tessio, the smart business choice was not realy a choice.

Stop trying to make it a personal matter. You have a better chance of turning water into wine.