It is pretty well known the Middletons all put their eggs in the basket with Kate and William, though. It wouldn't shock me Carole who was a known social climber and frankly obsessed with it through her children would push for it, but I think Kate had agency of her own too and wanted to go for it. I believe she and Meghan are similar that way - very steely, ruthless, will go any lengths for what they want. So Kate did. The opportunity was there, and the entire family capitalized on it. Paying for his holidays (he and Harry are notoriously cheap and would whinge to have their Mustique, Maldives, etc holidays covered and they were), ensconcing him in their family bubble as the #1, Kate turning a blind eye to his behaviors and disregard because of the great prize she'd be getting and Carole making sure they always got back together.
A lot of The Crown seems to be aimed, too, to American audiences who think Charles is some dastardly evil villain who murdered Saint Diana, and people who think Diana was a perfect halcyon example of humanity.