R514 - he's too lazy to be dissolute. What the Danes see him as as someone a bit like Prince Harry, only he got born the Heir rather than the Spare, so was forced to behave better. Mary came in and anchored him is the prevailing view, and worked harder than he do even whilst pregnant. She seems never to forget her incredible luck in landing him and the life. I don't agree he looks disengaged in the family photos, he looks just as much as he loves his kids as Mary does.
One thing you have to remember about Frederik and Joachim is that they had an emotionally disengaged Mum who was also their Queen - not unlike QEII's problems with her kids. That said, both Danish princes married well. Joachim's marriage ended in divorce but Alexandra was no Diana or Meghan - it was a much more ordinary split and was handled with a minimum of fuss. Joachim's second wife, just like Mary and Joachim's first wife, handle their roles well. Those sensible marriages more or less redeemed two princes one of whom was mostly interested in sports and found royal life boring (Frederik), the other of whom had a reputation for being testy with moderate interest in agriculture and military matters - but not on any level like the interest in environment that, say, Charles has in Britain.
The three Danish royal wives, including the ex-, have done a good job keeping the side up. Joachim has a lovely estate called Shackenborg; it would roughly be the equivalent of Frogmore Cottage for the Harkles, only it's much classier (there are photos of the inside available) and Marie, the second wife of the spare, has been only too happy to find herself ensconced there, and doesn't spend her time grousing about the huge apartments in Amalienborg Palace renovated for the Crown Prince and his family.
Frederik is now over fifty. He may not find his role wholly fulfilling, but he's acclimated to it and I don't doubt is happy with the brood Mary provided, and they are viewed as good, hands-on parents.
I can well imagine QEII looking at Margrethe's family and wondering why things couldn't be managed more like that, including the problems, on her end.
The Danish monarchy, by the way, is the oldest in Europe in straight lines of descent as well as sheer time. When royals from different countries are gathered, e.g., funerals, weddings, etc., they are seated and ranked by that. Queen Margrethe is always ahead of, say, Queen Mathilde of Belgium.