In 2004, John Kerry selecting as his running mate a man whose government experience amounted to a single term in the US Senate, which was not terribly remarkable, and if he had run for reelection that year he may have lost (the Democrat who Edwards endorsed to replace him lost). His time in the Senate was unremarkable. But he gave a good speech, checked the right demographic boxes, and the press liked him. He also had a small child and an appealing family narrative.
But I don’t recall anyone talking about how Edwards was manifestly unqualified to be a hearbeat away from the presidency, about how irrespondible John Kerry was for picking a VP candidate with so little experience. Though Kerry and Edwards lost, the selection of Edwards was way down on the list of reasons given, and even then it was the lack of chemistry between Kerry and Edwards.
Now, Edwards was a national figure because he had run for the presidential nomination. But I don’t think that’ a qualification, either, just that he was a familiar face.