I don't disagree with anything you said, but Jimmy Carter won the nomination and election in 1976 despite being a virtual unknown nationally when he started his campaign. Carter's entire political experience when he ran for POTUS was four years as an obscure state senator in Georgia, a failed run for governor in 1966, and a successful run for governor in 1970, where he served a single obscure term. Virtually no one nationally had heard of him when he announced his candidacy for the 1976 Democratic nomination. He won in large part because he turned out to be the perfect candidate for his time - in the aftermath of Watergate he had no connection to Washington nor was he seen as a party insider, and he ran as an outsider who was untainted by DC scandals, said he would never lie to the American people (as LBJ and Nixon had), and would be a fresh face on the national scene. He ran as much on his personal character and integrity as he did on the issues, which would also seem to be a strength of Beshear. He also had no foreign-policy experience at all, although he had served in the military in the Navy and had graduated from the Naval Academy.
Admittedly, Carter's single unhappy term as POTUS may only add to your point, but I do think that the liabilities you mentioned for Beshear may not hurt him at all, and his status as a DC outsider may well help him in 2028, especially after four exhausting and chaotic years of Dear Leader.