I think there must be a line in the sand at some point because if it was a private I don’t care if William is not there. Maybe one of the children or him was sick or Cate had a bad day. Andrew there could be stomach better i guess. If it was public both William absence and Andrew presence is problematic. What I’m trying to said in the end is that we always criticised the Queen for being an ostrich but in reality Charles is the same hiding behind his sickness (which frankly I understand the difficulty of battling with cancer as I have a parent that is experiencing it). But if he can travel up a down to Sadigram to London he can tell his brother to stay home and his son to pull it together.
I think they were trying a classic fudge* with the public/private thing. It being in the CC makes it an official event, but they didn’t list Andrew by name in the CC. I think that shows they know that strictly speaking Andrew should not have been there, much less led the family walk to the chapel. But I don’t think the BRF understands why people are disgusted with Andrew, and keep hoping it will all blow over. They are very much on his side, but trying not to anger the public (nonces are public enemy #1 in Britain and the public are very angry).
As far as Charles taking William to task, as soon as William became Prince of Wales Charles lost all his leverage. The system is set up so that the king can’t make his heir do whatever he wants, the heir has his own money, properties and position. So Charles can remonstrate with him but he has nothing to hold over William.
*this kind of fudge is very common in England it’s how we ended up with a state church that is both (anglo) catholic and protestant, for example.