I very much doubt it's the personal statement. I did admissions (for Oxford not Cambridge but similar) and it would be very difficult for the personal statement to say anything that would put us off. I haven't had time to completely read the thread but I'll tell you exactly what happened for us in Oxford.
Any students who didn't have the required predicted grades would be thrown out immediately (almost all had a straight sweep of A with a smattering of As at GCSE and in the new system would have mainly straight As prediction at A-level but as long as their predictions were at least at the level of the offer and the relevant GCSE subjects were at A* it shouldn't be a problem).
The main cut off would then be the pre-test (for physics and maths at Oxford this was taken prior to interview and would be a massive cut off). Even for students who were accepted for interview the pre test would be a massive part of the decisions as to whether they were accepted after interview as statistically it was the best predictor for degree classification.
Someone would look briefly at the reference (almost all schools write identical references claiming the student is the best ever so this isn't in the least helpful) and personal statement (again doesn't carry much weight most are the same - some international students have slightly poorer English or contain info that is less relevant but it isn't held against them).
In terms of which school you're coming from the person reviewing will not pay any attention to this - most will never have heard of 99% of schools anyway and won't know whether it's private or state. The exception of this is if the school has ever been put into special measures (this might not be the exact terminology it was a few years ago - essentially a school that was deemed to be failing) or if the student has ever been in care it is flagged up and the bar for getting an interview is significantly lowered.
I do think students from less academic schools have a disadvantage when it comes to the pretests and STEP papers which are taken at the end of the course. They are much more difficult papers and being at a school where the students are routinely stretched beyond the syllabus is enormously helpful.
You should be able to get the pre test results back and they usually publish the average result of candidates accepted for interview.