These are generally long-term arrangements, often for life, sometimes they even on to the next generation. An explicit fee might be fine this year, but what about next? In 10 years time? 50 years? How would changes be negotiated and how much would all that cost? An AUM based fee is agreed up front for the duration, is clear to all parties and gives the manager a direct incentive to increase the holdings.
I doubt too many managers would be willing to base an incentive on out-performance of peers, the Dow or S&P maybe...