I don't remember it that way at all. The opposite in fact. Charles would have been annihilated if he hadn't gone (and William and Harry would never have forgiven him. )
I do remember lots of 'outcry' as to why TRF weren't on public display weeping and wailing. There was talk about flags not being at half mast at Royal premises but the reason was that she was not Royal so why would they? It was protocol at the time.
I think if TQ and PP had of had their way, they'd have stayed in Scotland with the boys and Diana would have had a private, family funeral but again, the public demanded a state funeral. I don't remember any particular outcry at the time to the boys walking behind the coffin. It was all about 'oh those poor boys' one minute and then 'where are they? We want to see them?' the next.
I think it was really sad when it was either Harry or William said they were consoling members of the public over the death of their mother. Ludicrous.