I would really love to find something that would rid myself of privet permanantly. That is a bane to my existence.
I would suggest Ortho 'Weed-Be-Gone' in a sprayer, on the foliage
If perchance this isn't successful, Monsanto has chemical control:'Roundup', which will kill everything , but it does have possible residual soil issues,
if you wish to vegetable garden there.
That is why I would suggest you try the 'Weed-Be-Gone' first.
After cutting it down paint the exposed cut portion of the root with a “kill-it-all” weed killer. Cut it at an angle so there’s a lot of exposed cut area to paint.
Works for honeysuckle too.