It's generally accepted that you should use getter and setter functions for field access in an API and never allow directly reading or writing them.

Making types opaque (only interactable through pointers to them) simplifies the backwards compatibility story. Since 5.0, the internals of ALLEGRO_BITMAP changed significantly, and that would not have been possible if all the variables in that struct were directly accessible. I suppose that doesn't really apply to width/height, since those remained basically the same... but consider al_get_bitmap_flags. By your argument, you'd prefer bitmap->flags, and that was what al_get_bitmap_flags returned... but recently, it got changed to this:

int al_get_bitmap_flags(ALLEGRO_BITMAP *bitmap)
{
   if (bitmap->parent)
      return bitmap->parent->_flags;
   else
      return bitmap->_flags;
}

I.e. sub-bitmaps now inherit flags from the parent. Something like this would not have been possible to do in a backwards compatible way if we let you access that member directly.