Yes, all those things are crucial, and could explain the difference in results, surely? "Similar grades" is a very blunt measure, as others have pointed out, when it comes to the most competitive courses, and it is a weakness of the report that it does not specify beyond this. Of those applying for medicine, nearly all have "similar grades" - very high ones.
Subject choice, motivation, guidance, focus and so on, are all crucial, and the report provides evidence of differences between the sectors. And, as I keep saying, of one comprehensive where they addressed all this, and got excellent results.
There may be an unexplained bias, but none of the figures you quote demonstrate it, I don't think. There are real inequalities in the system, with sound data from lots of sources to show how they arise. IMO, these inequalities should be addressed, urgently and by raising taxes if it comes to it.
Other sources provide similar evidence:
www.ox.ac.uk/about_the_university/facts_and_figures/undergraduate_admissions_statistics/school_type.html.
(including the fact that 33% of those getting AAA at A level are from independent schools).
I live in London and the independent schools I have encountered do tend to provide feedback for children who narrowly miss the mark (often those who are interviewed), as well as encouragement to try again later if they wish. If someone is wide of the mark, I am not sure anything would help really - probably best to forget about it and move on! I seriously doubt anyone making the choice of sixth form would rely on comments about how good they were at fractions aged 10 to guide them - they will have a host of other results including public exams by then.