It's possible that Bellomo was waiting to get off supervised release to become boss. It's not an unheard of policy - Andy Russo waited for the terms of his parole to end before becoming the Colombo acting boss after Ralph DeLeo.

So, Danny Leo was acting boss from 2005 to 2007. In June 2009 it was reported that the family was using a rotating panel of leaders that had included Bellomo (released from prison in 2008), Venero Mangano (released from prison 2006), Ernest Muscarella (released from prison 2008) and Lawrence Dentico (released from prison 2009). In September 2009, it was reported that the family was using a three-man panel and that Tino Fiumara (released from prison in 2005) was on it. In October 2009, it was reported that Bellomo, Dentico, Muscarella and Dominick Cirillo (released from prison 2008) were all running the family to some extent. Fiumara died in 2010, and in 2014 an article cited the belief that the family was still using a panel.

In an interview over on Black Hand Forum, Gene Borello said that when he was last on the street (2014) the Genovese family was being run by Bellomo and Cirillo, though he didn't specify whether this was a panel of some other arrangement. In 2016, Jerry Capeci reported that Bellomo was the boss, and had been for a few years, and that Peter DiChiara was serving as his street boss. DiChiara was also identified as Bellomo's consigliere. Mangano died in 2017 and DiChiara died in 2018. After the murder of Frank Cali, a tape recording was made I think of Gambino capo Andrew Campos describing Muscarella as the underboss of the Genovese family.

In a recent Gang Land article, a LE source said that their information had Bellomo becoming boss around five years ago, which would put it at around 2015, 2014.

So, the Genovese admin as we currently know it is:

Boss - Liborio Bellomo, 63 (became boss c.2014)
Underboss - Ernest Muscarella, 76
Consigliere - unknown, Peter DiChiara died 2018
Street Boss - unknown