I think I finally "get" Popper. He has a mathematician's outlook: every statement has (at most) one meaning, which can be either true or false. The truth value is constant and uncontroversial: the same for all people, at any point in history. He is a good Cartesian in this respect.
Unfortunately, the consequence is that anything other than pure mathematics is either meaningless or false. Scientific theories may well be useful, but they are (or will be) falsified; that is all we can say about them. This is not to demean Popper's contribution, which is foundational, but the foundation by itself is not very useful. In particular, there's no place in his system for a new theory that improves upon an old one without merely displacing it.
The same tendency is apparent when he discusses philosophy (and philosophers) in The Open Society. Popper feels no empathy for Plato and does not try to "inhabit" his thought, nor does he try to place the old Greek in the historical context, unless to better condemn him, as in the final section of the first volume. I suppose that if a contemporary author were to hold now Plato's position we would criticize her for her social views (and decide she is a quite poor logician). Should we do the same with Plato himself?
Marx receives the same treatment. Contrast Popper's description with Isaiah Berlin's life of Marx, which shows much more historical and even personal insight, although Berlin was far from being a Marxist.