I am a Primary School Teacher and I have found this thread fascinating!
You are all absolutely right in saying a child should NEVER be forced to stay on a particular reading stage if they are ready to move on. I cannot believe there are so many schools/teachers who seem to be doing that.
It doesn't matter what level the other children in the class are reading at, this is irrelevant. Your child's class teacher should be able to tell you where your child is in relation to the National Average expectations, NOT in comparison to other children in the class. This way moving schools would not alter a child's progress.
Different book bands are probably stored in another area, in most schools classes do share books because of the wide range of abilities in most classes.
Sadly, the government expectations of what a class teacher has to do does take away the one to one reading time with the class teacher. But the way you have to think of this is that if a teacher reads for 5 mins with a class of 30, that is 5 minutes gained for your child, but 145 minutes lost each week (the equivalent of at least 2 whole lessons).
Also, reading is not just about sitting with a book. It is a combination of a range of skills that makes up the reading level.
Teachers read with groups during guided reading sessions so they can hear a child's fluency, identify any sounds that are unknown, identify which sounds they have consolidated, pick out strategies the child is using to decode words, asses children's ability to discuss a book (much easier for the child in a group with peers), assess comprehension skills and much much more.
Your child will do phonics lessons which helps them learn phonemes and graphemes needed for reading, literacy lessons where they will have a class text to read and do lessons which link to this, a school library where your child can choose a book to take home, and a reading corner in the classroom where they can read books freely.
All in all, it is not as important for a child to read to a teacher one to one as you might think.
Teachers might want to keep a child on a particular book band because they have assessed them at that level even though it sounds to you like they can read it perfectly.
If you are concerned that the teacher cannot explain to you why your child is not being moved up, then I suggest you go to the Head and put your case forward. Ask them to explain how your child is doing in relation to National expectations and, if above, how your child is being challenged.
A massive target for schools is to nurture the G&T (gifted and talented) NOT just the SEN (special educational needs) children.
Children do not "all move up together". Never have and never will.