You can't apply the rules differently to different types of state school, that wouldn't be fair.
Sure you can. if the rationale is that school that don't have an attendance problem don't need these rules, then state or otherwise that don't have this problem don't need these rules.
That document pretty much proves the link between absence and achievment for example:
Link? yes. Causation? Not one eensy weensy bit as far as I can tell (I only scanned the document, so if someone finds any controls, that would be good). In fact, the document is pretty explicit about this very big piece of the puzzle missing.
Absence is not the only factor that has a link to a pupil’s attainment. There are other complex relationships, such as the strong link from prior attainment and the link between different pupil characteristics and attainment, which should also be taken into account. This release aims only to provide an overview of the link between pupil absence and attainment based on available evidence.
So, frankly, I am not convinced it's worth the electrons it's using on the internet. What's entirely likely is the same child will attain the same amount if they miss any small amount. What of course it doesn't show is attainment and absence caused by the same factor (parents).