Trade-off between variable stability and discriminatory model quality
I think there might be a bit of confusion here: what is usually called a stable model is a model for which the performance doesn't vary (or not significantly) when sampling a different subset of training data or test data. In other words a model is stable if chance doesn't affect its performance. Typically one can use cross-validation in order to assess the stability of a model: if the variance is high between different splits then the model is unstable. Unstability is often a sign of overfitting, so one should be very cautious with an unstable model even if it seems to perform well: the risk is that the model might obtain good performance on the test set by chance, which means that it will actually perform poorly on new data but there will be no way to detect the problem. This is why people might favor a stable model to a high performance in some applications, because the consequences of deploying an unstable model in production would be costly.
But what you are talking about is not general model stability, it's stability of the performance across time for time-based data. That's a completely different question, it's about whether the model can represent time variations accurately. There are many types of models which can do that, but of course the more complex the model the more data it needs and the higher the risk of overfitting.