frogsoup, I'm also a sociologist by training. Didn't read the study itself, just the report; but from that, the very, very obvious explanation for the finding that deferred SBs don't do better than non-deferred SBs (and therefore do worse than Autumn-borns despite being older than them) is 'selection'.
Only those SBs defer who are so far 'behind' (in some sense or other) that even being a year older will only just allow them to keep up with 'regular' SBs; even being the oldest in the year, they will be significantly behind the average in their year.
This selection happens. Many LAs/HTs do not allow deferral unless there is professional evidence for exactly this kind of major delay.
If anything at all, the study shows us that where LAs do allow parents to choose, parents choose to defer (on average) only when the child is so far 'behind' that they will still be 'behind' (i.e. at the level of an average non-deferred SB) in the year below. They are a year 'behind' average SBs (who are 'behind' the average of their school year cohort themselves), rather than merely a year 'behind' the average of their school year cohort.
So if anything, the report is an indication that at age 3 (when these choices are made), parents really do know best; and that in our culture of overwhelmingly keeping children in their chronological cohort, there is no evidence for parents trying to obtain a relative advantage by 'redshirting'.
(I know this is a known phenomenon in the US, where children are taught out-of-cohort for all manner of reasons; but it seems that here, people are generally very, very cautious about pushing for either acceleration or decelaration. In a context where nearly everyone is taught within-cohort, being out-of-cohort has additional challenges and downsides; and it seems that people do take these into account before choosing to defer/pushing for acceleration.)
If at age 3, parents know best, then if you take cost out of the equation (by providing continuation of free nursery places for 4yos who defer), allowing deferral by parental choice might actually help to equalise socioeconomic differences.
As it is, in most places only those parents who are so much 'on the ball', engaged, informed, and educated that they were able to discern that their child was e.g. actually language delayed rather than just 'slow to talk', able to push for assessment very early, to avoid long waiting lists by going private, etc; will have a diagnosis or 'professional evidence' needed to achieve deferral.