I think the other thing to say is that reading books sent home are onkly a very tiny proportion of the 'reading provision' that a school provides.
For example, I teach reading in guided reading lessons, and i mark the children's books. We assess reading through standardised tests that are essentially comprehension exercises testing a wide range of skills against a range of texts. We may do online or on paper comprehension exercises as short activities throughout the year.
We teach phonics for reading and writing. We also do specific sessions with children who struggle with inference, using a particular resource.
We also read a class reader every day, and read a very wide range of texts in English lessons.
We visit the school library on rotation, and every child can havea library book and a class reading shelf book as well as their reading scheme book.
NONE of the above takes any account of, or uses, the children's individual reading scheme books in any way at all.
Yes, we also read individually and we assess children to move them through the reading scheme levels so that they have an appropriate book to take home. Daily reading at home - of something - is part of children's homework diet. However, a child's reading scheme level, and their individual reading book, is not used in in-school teaching of reading AT ALL, and failure to quickly move children up through reading bands may be in no way indicative of the deep, wide, far-reaching diet of books and reading teaching that the children are receiving in school.
It's like saying that a school's spelling homework sheets are routine and repetitive, which must be an indication that the teaching of English must be dull. There is virtually no correlation between the two.