Thank you for pointing out this important and hugely interesting paper. I strongly recommend that anyone who has an interest in the topic reads it. It has been cited 40 times in 3 - 4 years, including by the IFS, and achieved headlines in the Times, Telegraph and Daily Mail. If one is pressed for time (I am not - as you might surmise) and skims it, they will probably come away with the same impression as you. It is a perfect example of what passes for educational research and how it is received. There are countless red flags.
For a start, and to be clear, even the authors state that they do not view their results as "truly causal"; and "there is no private school advantage" for attending a Russell Group University. (My bold print, their words, although you need to read beyond the abstract for this.) My guess is that journal reviewers insisted on the first caveat.
To take your point about class size: The paper provides no evidence robust or otherwise that "small group teaching and a lot of support" are the explanation for the "highest results". The authors do however cite a reference that was certainly intended to give this impression when they say "The recent evidence from quasi-experimental studies find significant effects of resources on academic outcomes... including, specifically, modest but significant effects of lower class sizes (e.g. Fredriksson, Öckert, & Oosterbeek, 2013)" I can only assume that the expectation was that no one would read that paper because it concerned primary school children. As I think I said earlier in the thread, there is no evidence for a class-size effect at secondary.
Likewise, the authors cite Ndaji Little and Coe, and Smith-Woolley to emphasise the importance of a private schooling effect on attainment, when the former showed that "students’ prior ability was the single highest contributing factor to predicting the GCSE outcome of the models" and the latter that "genetic and exam differences between school types are primarily due to the heritable characteristics involved in pupil admission".
Arguably the single greatest deficiency of the paper is that the "value added" aspect of school type was based on the finding that a gap that existed at GCSE between private and state schools increased at A level. There all sorts of possible explanations for this, including the confounding effect of IGCSEs and the different characters of pre and post sixteen education.
It would be useful to know when the research was carried out, was it soon before 2020 using 10 year old data (ie from when Cambridge admission was meritocratic) or was it carried out around 2010 and it took 10 years to get it published?
The bottom line, if one accepts the paper at face value, is that no justification was found for RG universities discriminating against private school children. But that is not how the publication has been used.