Educational Psychologists use similar checks when assessing children with reading/literacy difficulties and it's also a feature of most (possibly all) Dyslexia screening programs.
Dyslexia screening programmes used in schools miss about 20% of children with dyslexia.
The standardised assessments used to assess for literacy difficulties also use "sight reading" as a measure; for instance, some of the most commonly used tests such as TOWRE-2 provides composite scores based on sight reading efficiency (how quickly high frequency words can be retrieved from memory) and phonemic decoding efficiency (how quickly non-words can be read) - the scores are combined to provide a measure of word reading efficiency. Obviously, this ability is closely linked to fluency, which in turn affects comprehension.
"the phonics screening check will find children who are not at the expected level, but not why. " it's a diagnostic tool and will identify gaps in knowledge and skills if used correctly by the school.*
This assumes that the information is used correctly by the school - unfortunately, as someone who visits/assesses children from many different schools, my experience suggests that it is often a tick box exercise.
Another test used in assessments is CTTOP-2 which measures phonological processing includes subtests which provide measures of phonological awareness - not to be confused with phonics. In recent years, assessors have found that (due to good phonics teaching) children often now achieving average scores for some phonological awareness sub-tests, but continue to have deficits in areas such as phonological memory and rapid naming both important for reading and writing. In short, good phonics teaching may mean that some children will pass the phonics screening test, but continue to have underlying difficulties that affect reading and writing.