In a for profit company, the point of the people evaluating your policy against needed care is essentially to find ways to deny you as much care as possible. In a universal system, the motivations aren't organized around whether a profit will be made or not and so the public (rather than private) bureaucrats will at least not be actively trying to kill or otherwise immiserate you. As an additional check on their behavior, the organization is directly accountable to elected representatives, very unlike the private system.
And Charlie Gard would have disagreed with Stephen had the court not decided that he should die.
There is another high profile ALS sufferer in the US: Jason Becker, a super virtuoso guitarist popular in the 80s, tragically afflicted at the very peak of his ability. Still alive, still writing music, but can’t move anything but his eyes or breathe. No one sentenced him to death or anything.