Thanks and I think it will. E6000 glue is very strong yet flexible after cured (72hours), the core of strap are glued to both top and bottom of leather at broken point, I even use needle head to push some glue into the hollow part of top portion of leather strap. Since the exposed core was coated with glue prior to pull down tight into bottom leather straw, that will make the broken strap whole again once cured properly. E6000 can be used to glue back the outsoles of shoes too and will hold very well. When it is leather on leather, once cured you won't be able to peel them apart again. The down side of E6000 can be difficult to work with, you only have a minute or 2 to adjust things, can not be disturbed while curing, and can be a little messy too. The excess is cleanable with acetone.
Since it is a tiny bag with tiny straps, I won't be put too much weight in it, otherwise other strap may break as well.
The core is suppose to add strength to the strap, but will be useless if it is not binded at the end to the leather. That was why the strap broke, the leather is very thin, and the core came out freely while I was cleaning the bag.