Tessio's murder was 'Family Business' and therefore probably not an issue that would need to be explored in the Senate Hearings.

As for 'ALL' Cicci's testimony, let's remember this is fiction not a documentary.

Apple

True. I reckon he began telling a story about the time Clemenza fell through the floor under his own weight

In addition to the benefits of cooperating with the police that you pointed out, I always believed that Cicci turned rat for one of the same reasons the Pentangelli did, Michael Corleone betrayed them. Remember at that point in the movie, Cicci was working for Pentangelli, not Michael. Pentangelli still reported to Michael so he wasn't really the Don, but he was in control of the Family's old NYC operations. As far as Cicci was concerned, Pentangelli was the boss and his boss has been betrayed. He didn't hear the, "Michael Corleone says hello" line, but Michael sent Pentangelli and him to meet the Rosattos and the Rosattos tried to kill them both. Even without hearing the famous line, it would still look like Michael set them up.

Or The Rosato's just went against what they said they'd do. Cicci would have to be slicker then your average to know it was Michael without proof