I’m updating this with the (unrelated) cause of this problem. I traced the call to the webWorkerPromise called here that was never being returned. That led me to look into the itk.js setup more carefully and I realized that I had not configured webpack correctly per instructions here so that the WebWorkers were available. However, since my app is based on Create React App, the webpack configs were not immediately changeable and I ended up using craco-itk so that I could use the WebWorkers required for reading in the .nii format files. I will also mention that there is Create ITK App for bootstraping a React App, though it does use CRACO, so it’s not “100% CRA” but also not ejected.
All in all, the vtkITKImageReader was a bit of a red-herring but hopefully this can help someone in the future.