I certainly wasn't "thrilled" that Charles said he's proud of Katherine. I find the whole concept of pride in someone else a bit odd at the best of times. What's the point of feeling proud of someone else, and how is it even possible? I have children. When they do something extraordinary I tell them that they should be proud of themselves - they worked for whatever it was, they should reap the rewards which for the main part should be a sense of pride that they achieved what they set out to do. I don't feel any pride on their behalf - it's their achievement, why would I appropriate it for myself? I'm delighted for them (thrilled, even!), so chuffed to see them happy, I'm comforted and reassured that they're growing well - but not pride. That "I did it, yessss!!" feeling is theirs, not mine. That's how I feel about my own achievements: nobody can take them away from me, I did it, I'm proud of myself.
That said, what's touching about a father-in-law saying such a thing about a daughter-in-law is the public declaration of the admiration he feels for whatever it is he said he's proud of. And, given he's the King, it's also a public warning: she has my stamp of approval. From what I saw in the papers, he didn't say he's proud of her for dealing with the cancer diagnosis (it's not like she's got any choice, really). I read it as admiring of her choice to make a public statement, that public statement, the way she did. There are very few people in the world who know what it's like to be under a microscope like any incumbent Princess of Wales. They will have been aware of the social media frenzy about her whereabouts. I think he was expressing admiration for the way in which she handled that noise, whilst also dealing with a cancer diagnosis and treatment, whilst also trying to protect her public-facing DC from unwanted input, whilst also facing the not-unrealistic prospect that she and her husband might become King and Queen earlier than they anticipated. By any measure, that's a lot, isn't it?