I seem to have found a workaround by exporting my sequence as XML, then using Import > "Pro Import After Effects..." in After Effects. I'm still confused as to why it doesn't work the normal way though.
Try the method of right-clicking on the clip in Premiere and selecting "Replace with After Effects Composition" instead.
I am a little confused by your explanation. Are you telling us that there is no master comp with all of the shots (clips) from the Premiere Pro sequence? There is nothing keeping you from Motion Stabilizing or Warp Stabilizing a nested comp. I Warp Stabilize and track nested comps all the time. I am guessing that you want to Warp Stabilize rather than Motion Stabilize.
I am also guessing that you are not using Dynamic Link. If your are trying to use Dynamic Link to Warp Stabilize a bunch of shots from a Premiere Pro sequence then I would suggest that you render and replace any of the shots that are longer than two or three seconds after applying Warp Stabilize in AE because previews and rendering your final movie inside Premiere Pro when you have a lot of Warp Stabilized footage is going to take a very long time and put way more stress on system resources than pre-rendering the processed footage to a DI (digital intermediate with a suitable frame based production codec).
Please clarify your workflow and your OS, AE, and PPro versions down to the last decimal point.
I found out what caused this. I had done a quick grade via Direct Link to Speedgrade, continued editing that sequence in Premiere Pro, then tried all the different ways of bringing that sequence into After Effects, trying both Direct Link and replace with After Effects Composition, and importing the Premiere Project into After Effects.
When I remembered that I had used Speedgrade on that sequence, I opened an older sequence in Premiere Pro that hadn't been linked with Speedgrade. Opening that older sequence in After Effects using any method worked as expected (clips were in the composition as clips, not each clip as a comp within the comp).
Is there a way to remove that direct link to Speedgrade, so I can then proceed with After Effects and grade as the final step?