If they knew that Harry was always rather shaky mentally, (and since this book I am wondering if he was like this since childhood?) I wonder if they worried a lot about letting him go to the US and start off life there? DID they fear that he would become even more unstable in a strange environment, without staff to keep an eye on him, and with a wife who certainly did not encourage his better side? I suppose they felt there was nothing much they could do...If he was determined to make money outside the RF, he was a grown man and if he wasn't visibly acting strangely or harming himself there was nothing much they couild do.
In my opinion his family is worried about him. However like you stated Denville, unfortunately there's little that they can do. Harry's already shared about his in person conversation a with William and Charles after the DoE's funeral. A conversation that appears to have been very emotionally charged. So the impression that I had after reading about it, is that his father and brother were very concerned.