Ok, I think I've got the proof I need. Strickland's Lemma 3.7 requires a compactness condition: for every compact $C$ in $Y$, $C\cap X = C\cap X_\xi$ for some $\xi$. I can prove this condition when each $X_\xi \to X_{\xi+1}$ is obtained by a a cone attachment. I'll post details when I've got them written up.

@DylanWilson Not in the category of pointed spaces!

@MattFeller I see your point, but I don't think these inclusions can be thought of as cone attachments.

Conversely, I think Lemma 3.7 in Strickland's notes may do the job for me. I'm checking details before suggesting Dylan Wilson upgrade to an answer, though.

The union gets the subspace topology.

@TimothyChow whenever an intuition like this fails, the natural next step is to ask how badly it fails. So: “Ok, homotopy groups are not homology groups; what properties do they have in common? How can we compare them?”

Perhaps homology? The folk stories that I've heard about the early days of the homotopy groups have to do with expectations being set by what was known about homology.