The blurry texture is exactly as Sir Teatei Moonlight described, MTL cannot convey the "Closest" texture setting. There are also vertex colors that should be multiplied with the texture that cannot be conveyed by MTL (but Blender does support exporting/importing vertex colors in OBJ since 3.3).
Normally for submitting people just make sure the texture is in the diffuse slot (map_Kd) and don't worry about all the stuff that can't be conveyed. So it would be just like you had in your first post.
Normally for submitting people just make sure the texture is in the diffuse slot (map_Kd) and don't worry about all the stuff that can't be conveyed. So it would be just like you had in your first post.