Correct. They get paid what they do because they set their own compensation. Performance has nothing to do with it, unless things get so bad they are removed.
True, but also a myopic understanding of what actually goes on. Who do you think actually serve on the boards? The answer is other executives. And so there's the all too common situation where pay raises and golden parachutes are essentially quid pro quos.
That situation would require 2 people to have alternate roles at the same companies which is incredibly rare, if not already prevented in the corporate bylaws.
Many board members are in entirely different industries and sectors and some only serve exclusively on boards. Many others are investors whose money is being used to pay those executives. There's also a layer of shareholders above the board and voting is required for all major decisions.
Board members have a portfolio of firms they are on, so they are less exposed to any single firm than staff. They are also part of the group that might be named CEO if a new one is needed, so it's in their interest to keep compensation high.
The board and the CEO are people who have known each other for years.
On the other hand, I don't think that is the only source of problem. Companies have gotten way too big and the experience of running global conglomerates is simply not common. Add risk aversion to it and the same people, whether succeeding or failing, end up getting hired over and over. It's a lack of supply and lack of competition.