There are several questions floating around this interesting thread:
First, as Olivant said, the hearing was a Senate investigation, not a grand jury; and in the pre-RICO days, it was not, strictly speaking, against the law to be a consigliere. In fact, unless a specific charge was leveled against the Don or a member of his family, it wasn't against the law to be in the Mafia. In fact, there was no way to stop any lawyer from representing an accused person in any kind of proceeding unless the lawyer had been disbarred or convicted of a crime.
Second, putting Tom's photo in the chart as consigliere was an attempt by the committee to show his connection to Michael and his criminal enterprise--no doubt about it. But, if you look at the photo of the chart on p. 216 of Lebo's book, you'll see that there are no footnotes under Hagen's name connecting him to specific criminal activities, such as shylocking, drugs, criminally receiving, etc.--just a FBI file #. Presumably, if Michael had gone down on the perjury charges, maybe the committee and the federal prosecutors would have started working on Tom, Fredo, Rocco, Neri, etc. But Michael was the target at that point.
Third, Cicci had a choice: to remain silent and honor omerta, or do what he did, which was to break omerta. He told things about the family that were supposed to be secret. Even though he didn't directly implicate Michael ("I never talked to him"), he said that Michael was "the head of the family." That one phrase in itself could have damaged Michael's legitimate front.