The MTL-File is fine (texture paths correct and no spaces in any names or paths) but when viewing the OBJ-File there are multiple faces appearing white when they should have texture data on them.
All of the faces are almost certainly being correctly ‘textured’, however back-faces’ materials are generally ignored by exporters and apps opening files like your.
If you reverse the faces, so that you no longer see their backs, applying fresh materials as necessary to the now visible faces, then these should export as you hope.
It is doing exporting and displaying it correctly - but sadly you have modeled it incorrectly, and need to do more work to fix things…
It’s unfortunate that authors of models like this can’t be bothered to orient faces correctly before texturing. A good lesson though for your own models. Stay on top of correct face orientation and fix them before you apply textures. It’ll save you or someone else unnecessary work later.