Yes, people certainly care about more than money. Economists have always understood this because the root concept has been not money but “utility.”

Naturally utility is hard to measure. As a first step, we can use a rational agent model and use money as a proxy for utility. The models can tell us a lot about bargaining and free trade. An example: if the price of apples falls, what might you expect? If you assume people are selfish, then you’d expect them to buy more. This is the law of demand which has vast application. But what if the assumption of selfishness is wrong? Notably the law of demand will still hold. Someone altruistic will buy more apples so they can feed more people. This example showed me why economics can be powerful and useful.

Nonetheless, there are biases where we systematically stray from rational predictions. Our models are wrong so we have to improve them. We now have the ability to look inside people’s brain, to an extent, and understand more of how we determine utility.

In the ultimatum game, we find that different cultures accept different offers. Some cultures will accept lowball offers while others find them offensive. These experiments tell us about the beliefs of agents.

In summary, I still think traditional economics can teach us a lot, even if there are flaws.