#1 I agree that prize-based research only works in certain areas. That is why I would make it only one of the three ways to fund research. I think that any competition should be judged by outsiders, if the competition cannot be accurately judged by outsiders then it is not a good competition. For instance, a self driving car race can be judged by outsiders.
For #3 the accountability comes from the board of directors. Right now the highest ranking professors are basically accountable to no one. If a field starts to stagnate, get to esoteric, or close off to new ideas, there is no one with the power to get it unstuck. The top professors do not have an incentive to get it unstuck, because they are the most successful in the status quo, their status is directly linked to the current direction of the field. An independent board of directors has no such perverse incentive. Also, each institution would be actually independent, unlike with the grant system in which grants are centrally allocated by a network of the existing professors. So if one institution goes the wrong direction, another institution might try and make its name by trying out a novel area of research.
For #3 the accountability comes from the board of directors. Right now the highest ranking professors are basically accountable to no one. If a field starts to stagnate, get to esoteric, or close off to new ideas, there is no one with the power to get it unstuck. The top professors do not have an incentive to get it unstuck, because they are the most successful in the status quo, their status is directly linked to the current direction of the field. An independent board of directors has no such perverse incentive. Also, each institution would be actually independent, unlike with the grant system in which grants are centrally allocated by a network of the existing professors. So if one institution goes the wrong direction, another institution might try and make its name by trying out a novel area of research.