Note: I want to avoid modifying the model in the javascript code and do all the model design inside Blender.
Note #2: While this qu
The problem is you have not set up UV coordinates for your model. By default, each face applies the whole texture, but in blender the UVs are blank when exporting.
You want to specifically set up your UV coordinates. These are coordinates that show how to apply a texture to each face.
Make sure to UV unwrap your model in blender. Go to edit mode (tab), select all faces, press "u", and click "unwrap". Then try to re-export.
Unwrap is just 1 method, there are many. Experiment with different methods in blender to get the results you want (possibly the "reset" option).