How can I argue that detente was more effective in ending the Cold War than brinkmanship?
If you are going to make this argument, the best way to do so is to point out that the Cold War did not end during the time in which brinkmanship was most common. The Cold War did not end after the Cuban Missile Crisis or any other such confrontation. Instead, it ended at a time when President Reagan and Mikhail Gorbachev were on good terms with one another and were meeting on issues like arms reduction.
The mechanism that you could suggest is that detente allowed each side to be less afraid of the other. You can argue that the Soviet Union would never have been willing to try to undertake any reforms before detente. You can say that they would have been too afraid that the US would take advantage of what would seem like weakness to attack. By this thinking, the rise to power of a moderate reformer like Gorbachev would not have been possible before detente. In this way, detente was more effective than brinkmanship in ending the Cold War.