It is a matter of preference to the braider on this matter. Me I don't like to use stopper knots in the way that has been done on that lead in your post above. This, I believe will cause problems later on no matter what kind of support you use as the core. To me, this way of braiding a lead compromises the braids integrity, not to mention I just plain don't like the way it looks. This is just my opinion and I am not speaking for others on that issue. Some prefer to do this simply because the beads hole is not large enough.
Another big improvement would be the choice in your leather. I don't know if you are using spool bought kangaroo, or some other leather. But I do have this to say. " There's nothing like using beveled lace to give that finished and professional look to your lead."
Based on what I do see in your braid itself, I would say that the braid is tight enough. The thing with braiding is consistently doing the braid the same all the way through the braid. Same tension equals a nice straight braid. You back braiding looks good from the photo. You will have to cover that with some type of knot in order to give a finished look.
The best way that I have found to braid show leads is to braid out the 36 inches with core or not with core. Depends on the braider. Then do your loops for the handle and snap end.
Brian...