Does this mean that if you’re still on Ashrei when the chazzan begins half-Kaddish, you stop and wait for him to finish (without answering)?
Why “without answering”? On the contrary, you definitely wait and answer. Even in psukei dezimra you answer kaddish (until da’amiron be’olmo); but in the second ashrei of shachris, or the ashrei of mincha, there is no inyan at all not to interrupt, so of course you answer for the whole kaddish, as well as greeting people and returning their greetings, etc.