Is there an intuitive interpretation of precision always higher than recall?
Generally lower recall means that the system is too strict, i.e. it predicts an instance as positive only when it has clear indications in the features that it's indeed a positive. As a consequence, it misses the true positive instances for which the indications are not so clear.
But when looking at macro-recall it's more complex: it depends primarily on how many classes there are and their distribution. In case of imbalance a very common problem is that the system predicts the majority class too often, causing a low recall for the minority class(es) and therefore a lower macro-recall. The only way to check for this is to look at the performance of individual classes.