Disclaimer: My academic coming of age was in an environment where behavioral economics played only a minor role. My research is theoretical, both "behavioral and non-behavioral", economcis.
I believe it is incorrect to say that game theorists (or economists) in general are not convinced by behavioral economics. You can easily see it is considered to be quite relevant by finding Thaler, Shiller, Kahneman and Selten (yes!) in this list. I also do not feel like young economists stay away from behavioral economics.
That being said, there are some (not necessarily good) reasons why many economists have a certain dislike for behavioral elements. One is that students often criticize standard models, taking a "behavioral approach" that in the end simply breaks down to "why should I study Bayesian Nash equilibrium if people are not clever enough to make such calculations anyway" -- this is a fair point, but if you want to have game-theoretic analysis or predictions you should base these on some closed system. I don't find "people are stupid and hence anything can happen" particularly illuminating. Luckily, this is not what modern behavioral economics does although some critics of "standard economics" seem to think this. Adding behavioral aspects often makes a model more complicated to analyze, which the same kind of critics then complain about again.
Behavioral economics often simply incorporates systematic "errors" (?) such as time inconsistency, loss aversion, social preferences, overconfidence, probability weighting and so on. This is reasonable because there is almost undeniable evidence of such biases.
An issue that I sometimes have with such models is that they don't deliver surprising insights. For instance, if you assume that a decision maker is overconfident and then find that this decision maker has a preference for riskier prospects than a "rational" decision maker without this bias, I am not very moved. The result just reflects the assumption. When I see the magician putting the rabbit in the hat, I don't clap when he pulls it out. I am not saying that every publishable result needs to be surprising, but it needs to add to our current knowledge. Serious economists know the limits of their models and are aware that changing the beliefs in some direction alters behavior.
By far not every behavioral paper is like that: many, in fact, expand our current knowledge and our theoretical toolbox tremendously. However, we then have many little toy models that are applicable in only some environments and not so much in others. In contrast, the rational benchmark is always applicable and usually comes with a normative touch: this is how utility maximizing agents are supposed to behave. Maybe it is just me, but I sometimes feel that this is also a point that many critics of "standard assumptions" actually mean: people should be more social and less egoistic and firms must not only maximize profits. That is, this normative aspect gets mixed with the analysis. However, you cannot blame behavioral economics for people having wrong perceptions about it.
Moreover, you usually cannot exploit homo oeconomicus. In contrast, the behavioral agents have systematic flaws by design and therefore they will be exploited. Behavioral agents often introduce a Dutch book. As a result, behavioral assumptions often only make sense in decision theory (i.e., individual decisions without interaction), but not so much in game theory (i.e, strategic interaction) because no equilibrium exists when a money pump is present. I guess what I am trying to say is that while some behavioral biases are reasonable and hard to deny, they -- if all their logical consequences are taken into account -- can still produce unreasonable predictions.A nice paper making such a point is the AER paper "Until the Bitter End" by Ebert & Strack discussed here.