Just to summarize:
There is absolutely no play in the headset now that the steerer is 2mm below the top of the stem and the top cap bolt can properly pull up on the steerer.
The two 2mm collars/shims that fit above (part names are from the Specialized "Roubaix Frame Installation Guide" pdf) the "headset cap/spacer" and below the stem are smooth and flush on both surfaces so I don't think there is an up/down issue.
I suspect that the adjustment shim/flange collar that fits inside the stem and around the steerer might have been slightly cocked or had a piece of grit under the flange and that made it just a part of a mm too high, thus preventing the top cap from engaging the stem.
Everything is A-OK now. When I do future stem angle adjustments I'll be sure to check that the stem is still above the steerer.