Game Theory seems sort of useless in the real world because people are not rational players, and the real challenge is in getting an accurate model of their behavior. The honor system would work probably fine in a tiny close-knit liberal arts college, while it would obviously wouldn't in a place where the degree itself is the target.
Game theory seemed kind of useful when the US was negotiating nuclear weapons control with the Soviets. It allowed successful negotiations in an extremely low trust situation.
Also, your own example is an application of game theory; you've basically stated a 'prisoner's dilemma' problem. You state that in a high trust society, most people will choose cooperation, while in a low trust society, most people will defect.
Aside from evolutionary biology, cancer research, embryonic development, economics, internet routing, spectrum auctions, counter-terrorism, kidney exchanges, generative AI, and preventing nuclear apocalypse...What HAS game theory done for us?!
Proved that everyone asking that question is playing a signalling game!