I feel like I'm able to give even a slightly valid response to this, as I've attended both a private school and a state school.
No, they should not have different exams. The school I used to attend is the private school- their grades are really quite low. I mean, when I left after doing my 13+ Common Entrance in Year 8, yes, I got 3 As and a few Bs, but that was the highest out of the three of us. The school in itself isn't doing amazingly; of course, I know in no way that that reflects off of the private school system as a whole.
However... my current school has a sixth form. We got 25% A*/A in the iGCSE English Language in November. Over 85% of students in the GCSE cohort last year got A*-C in at least 9 subjects, and this year, everyone on average is studying 13-15 subjects. In the sixth form, there were 5 Oxbridge applicants last year, 4 of which attended the school itself; 2 of these candidates got A*A*A* in their A Levels, and both went to Oxford and Cambridge to study Medicine respectively.
Of course, these are just small segments of data. Overall though, my point is that the amount of work you put in, as both an individual and a school, will reflect what you get out. Our school offer constant 'turbo sessions' as a way of ensuring we do well in exams, and since they started that A* grades at GCSE have risen by 24% [or so]. Other schools- such as the private and grammar schools my two friends go to- use 'study leave', and they're doing much worse than they anticipated.
This alone should explain to you that there is not a need for private/grammar/independent schools to be given a different treatment than state/senior/mainstream schools. Someone attending Eton can do just as well as someone attending one of the 329 failing secondary schools, as long as they put the effort in. I'm fully happy for my Latin exam to be compared to someone from Eton- after all, we're both humans, and both put in a relative amount of work.