There are boarding schools that accept very young children.
There are children who are sent to boarding school because they are inconvenient.
This does not mean they all do or they all are.
The Archers is not a public information broadcast on behalf of the Headmaster's and Headmistress' Conference!
Bertrand, of course not. But TA is supposed to be a realistic soap. They take great pride in the awards they won for various mental health storylines, e.g. Jack's Alzheimer's. In those cases, having gone to some lengths to find out what was typical, they tried to stick to that, broadly.
But on this key aspect of a major storyline they've either done no research at all or they ignored what they found out, because they wanted to pretend that Dotheboys Hall lives. It's a jarring note in what otherwise appears to have been a well-researched and realised storyline.
I'm not saying this because I'm a great advocate of boarding or independent schools. I've been equally critical of their inability to get other stuff right, often involving state schools. And they certainly don't get all the health stuff right, either. Helen's pre-eclampsia last time was a case in point - everybody very worried for a few hours, then suddenly everybody behaving as if the C section had sorted it all out, when in fact she would have been intensively monitored for days and days afterwards.