Sometimes they do, but sometimes those claims are not credible -- i.e. a group will have been invented out of whole cloth, or some other group will want to take credit for something that it had nothing to do with.  Any statements and/or videos claiming responsibility would have to be evaluated on their merits.

As for the investigation... well, that assumes the integrity of the officials involved, doesn't it?  If people believe that the state could have been involved, do you think they're going to believe what the state's own forensic investigators say about who did it?  Or believe the FBI's or Scotland Yard's investigators, if they were to be invited in?

In Lebanon, this is why they appointed the UN investigator into the Hariri assassination, and why they had several different forensic teams from different countries inspecting the evidence; but that also can prove problematic in different ways.