I found it interesting that the hosts of the podcast were speculating about Vernon Dursley's secret magical abilities when, in fact, Vernon is the only member of the Dursley household who is not magical in any respect at all. As we learn later in the series, both Petunia and Dudley do have a single, derivative magical ability: because they are blood-related to Lily, they have the ability to maintain, and also to break, the magical love-protection for Harry. (There is surely some sense in which maintaining the love-protection counts as a magical ability. After all, what is magical ability except the ability to affect magical forces and properties in the world?) And in fact, Petunia actually exercises and uses her single magical ability every day that she allows Harry to call her house his home (at least until he is of age). Petunia is undeniably the executor of Lily's love magic after her death, and I'm sure Dudley's blood relation to Lily had some effect on anchoring and maintaining the love magic as well. From this perspective, Vernon is decidedly a non-magical outsider in his own family.
I've often wondered whether Petunia understood that she really does have this one magical ability after Lily's death. I can imagine two possible effects that this magical ability might have had on Petunia's character.
One possibility is that Petunia did not understand that she was the executor of Lily's love magic after her death -- she didn't understand that Lily's death in fact gave her this one, derivative magical power or ability. In this case, I wonder whether Petunia might have had a better relationship with Harry if someone (Dumbledore?) had taken the time to explain this fact to her. For example, if we think that some of Petunia's abuse of Harry is partially rooted in her jealousy of Lily's magical ability, we might wonder: Would Petunia have treated Harry better if she properly understood her own magical role? With Lily's death, Petunia finally has an opportunity to participate in the magical world -- something she desperately wanted to do as a child. We might even think of Petunia's derivative magical ability as a final, unintentional gift from Lily: in death, Lily was able, in a very small way, to give Petunia some of the magic that drove them apart when they were growing up. In any case, perhaps if Petunia had fully understood that she was maintaining a magical effect and playing a crucial part in the magical world by taking Harry in, then some of her jealousy of Lily would have lessened, and Harry's upbringing might have looked very different.
The other possibility is that Petunia fully understood that by taking Harry in, she was introducing magic into her life and her son's in a profound way -- not just by taking in a magical child, but by accepting the magical burden of maintaining (instead of breaking) the love magic within the family bloodline. In this case, perhaps Petunia's awareness of this magical burden only added to her resentment of Lily and worsened her treatment of Harry. Perhaps Petunia's awareness that she now has one magical ability made her all the more angry at her sister: not only is she jealous of Lily's magical abilities, but now her sister has imposed a magical ability on her (Petunia), and what's worse, this imposed magical ability is not even really Petunia's own ability, but only an indirect, very powerful secondary effect from the talented sister she was so jealous of. In other words, Petunia is now as close to being magical as it is possible for a Muggle to be, but she still can't go to Hogwarts. She still doesn't have any intrinsic magical abilities. She still isn't as special as Lily (from her own perspective). And maybe this only added to Petunia's bitterness.
Just some thoughts.