A good 2005 article by Lois K. Fuller, "The 'Genitive Absolute' in New Testament/Hellenistic Greek: A Proposal for Clearer Understanding" makes the case that the so-called Genitive Absolute is a genitive construction for bringing an item of information as a piece of necessary prior knowledge essential to the whole understanding of the sentence, paragraph or discourse.
This makes complete sense to me and modifies the usual discussion of the Genitive Absolute as a construction using a participle and a noun or pronoun in the genitive not directly related to the subject of the sentence.
For example, Mark 14:33, "Immediately, while he was still speaking, Judas, one of the twelve, arrived; and with him there was a crowd with swords and clubs, from the chief priests, the scribes, and the elders" uses a genitive construction "while he was still speaking" which could be regarded as inessential to the arrival of Judas who is about to hand Jesus over. But in Gethsemane, Jesus in prayer to God wrestles with the dreadful challenge he is facing. In the next passage, Jesus' arrest presents a narrative tension between Jesus as acted upon (arrested) and as actor or agent in the proceedings (as he has been in Gethsemane). Mark 14:33 makes this narratively clear because the genitive construction identifies Jesus as still speaking at the time that something else happens, namely, the attempt to arrest him.
Mark 5:35 is another example of this construction: "While he was still speaking, some people came from the leader’s house to say, “Your daughter is dead. Why trouble the teacher any further?” In this episode, Jesus' reading of the situation prevails: "But overhearing what they said, Jesus said to the leader of the synagogue, “Do not fear, only believe.” And he describes the child as not dead but sleeping.
In these two examples, the genitive absolutes keep the counter point of Jesus' perspective in Mark's gospel alive. They offer a voice for the reader to hear against a cacophony of other voices.