When I took my 16+ (then called "O" levels)and 18+ "A" level examinations in the 1950s they were set by the rather grandly named Northern Universities Joint Matriculation Board, NUJMB or JMB for short. No one doubted their integrity or veracity, which may be because that was a more deferential age, but I suspect also because they did what they did honestly and to the best of their ability in the light of the teaching materials and examination techniques available at the time.
I don't actually know, but am fairly certain that, although they would levy a charge to cover their expensivenesses, they would aim to cover their costs rather than make a profit out of the enterprise. There were other examination boards run by other universities, notably London and Oxford and Cambridge, which covered other parts of the country and presumably had a similar ethic.
However, these university dominated boards have, over the years, been merged, amalgamated and I suspect, semi-privatised. If they don't actually make vast profits I suspect they pay there executives vast salaries based on the proportion of the "market" they manage to grab.
What Michael Gove's move, to restore the dominance of the elite universities, demonstrates, is that there are vast areas of human activity in which business ethics and the profit motive are inappropriate. Education is one of them. Would that he also recognised the malign effects of the gradual privatisation of other vast swathes of the education service, so that it is in danger of rapidly ceasing to be a service and becoming a business. See Melissa Benn's article in Open Democracy for details.
It would, in my view, be a mistake to place the posher universities in the driving seat for "A" level examinations, since they are, or should be, designed for much more that selection for universities. Those engaged in the actual teaching at "A" level should obviously have an important input, as they know more about it than anyone else.
Whatever new structure is devised should be firmly dedicated to stimulating the interests,enthusiasms and critical faculties of those who are going to spend two years of their lives to studying the resultant courses. Other issues, including the the requirements of the universities and the needs of business, should be secondary. The profits of the organisers should not feature at all.