....state.gov servers must have logs of all emails...
The problem is, that Hillary’s private server was a completely separate operating system. She had her own email accounts and addresses, and she only shared them with her most trusted friends. It was a closed loop, with no connection with the State Dept .gov system.
The only way to break in for any records would be to seize the computers and phones of all her contacts and see if you can work backwards to uncover any traces. It doesn’t seem that the FBI is going after these people with a great deal of urgency, allowing them to cover their tracks.