If anybody's reasoning is unsound it is yours.
To use emotive and inaccurate terms like 'murder' for a difficult case like this helps nobody.
You are effectively accusing the medical team of an unlawful act when this is clearly not the case.
It is also deeply unfair to the medics who not only had to care for Archie as best they could, knowing that his recovery was impossible, but also had to await the result of a lengthy legal process.
Keeping Archie alive effectively as an appendage to a machine is deeply dehumanising.
The outcome is based on compassion both for Archie and, in the end for his family, as it does not prolong their agony unnecessarily.