This is unlikely to be a satisfying answer but I don't anticipate a better one coming.
In another answer I have addressed the historicity of the Mahābhārata: Can the events of the Ramayana and Mahabharata be astronomically dated to around 7300 BCE and 5561 BCE respectively? There, I explored some of the problems of the Mahābhārata as oral literature. Namely, it is presumed that it was passed down for many generations as an oral epic before it was re-edited and written down. Anthropologists have observed that contrary to popular belief, oral cultures actually tend to preserve stories better and have longer, more unchanging memories than literary cultures. (The main academic source for this is Walter Ong, Orality and Literacy. There is also a more poetic, shorter, and freely available essay by Ananda Coomaraswamy: "The Bugbear of Literacy")
However, this tells us absolutely nothing about whether the characters in the Mahābhārata, including Krishna, actually lived. The traditional assumption in India has been basically ahistoric: that Krishna was a real person, but that he lives primarily through the telling of his story (i.e., what the ancient Greeks would have called ἱστορία historía) and in his mystical apparition to Hindu believers. The Western conception of "scientifically verifiable" history has come into conflict with the assumptions of many cultures, including Japan and India.
We do not have any archaeological evidence for Krishna himself, since India did not have a native tradition of sculpture; this was brought by the Greeks in the final centuries BC. We do have disputed ruins of the city of Dwarka. As you note in the question, these have been supposedly dated to 3000 BC. but many academics disagree with this.
After Dwarka, the oldest evidence of the name "Krishna" is in the Rig Veda, a more rigorously preserved set of oral prayers which has been dated to roughly 1500 BC based on linguistic evidence. This is merely used as a name and we cannot confirm who it is referring to. The oldest reliably dated references to parts of the Krishna story are the Chandogya Upanishad (500s BC) and Panini's famous Sanskrit grammar (300s BC). There is also a disputed reference to Krishna by a Greek author writing in 300 BC. (source: Edwin Francis Bryant's introduction to Krishna: A Sourcebook, Oxford University Press)
Here we see two things: (1) there is a 3000-year gap between Krishna's supposed birthdate and the earliest archaeologically verifiable references to him, but (2) before 500 BC, India was an oral culture without a sculptural tradition, so we cannot expect any much archaeological evidence for specific historical figures from that era anyway. For comparison, there is a 400-year gap between the earliest Buddhist scriptures and the historical dates for the Buddha, who most scholars believe existed, and a 80-year gap between outsider references to Jesus and his historical dates-- again, most scholars believe he existed, although some people try to put this in dispute.
In conclusion, there is unlikely to ever be as definite a proof for Krishna as there is for Jesus (or, say, Socrates or Caesar), but this simply reflects the respective times and cultures in which these people are said to have lived, and doesn't meet the generally high requirements for an argument from silence.
