> But Italian doctors believed different treatments were warranted
Again no.
"The Italian hospital had acknowledged it could not find a cure, but had proposed maintaining Alfie’s life for about two weeks while doctors tried to investigate his condition."
There are no treatments - even that seems to be agreed.
> In a private healthcare system, the parents would be free to pursue that
If they could afford to write a blank cheque for open ended life support. Any insurance would defer to the consensus of medical opinion and cease treatment. They would have ceased cover long before the 2 year trail of court cases and appeals all the way up to rejection from the European Court of Human Rights (not part of the EU or UK govt).