Ok, so the best thing is likely 2 cuts then. 1 as high as you can reach, and then 1 lower to the ground. Pull off the middle section. This will at least remove contact from humans/horses/etc.
Paint the one on the ground right after you cut it. that will work on killing the root system but yes, some suckers may survive, but they’ll be a lot easier to pull. For easy pulling I put on 2 pairs of disposable gloves (or even some heavy duty gloves that can be properly washed) and gently pull, being careful not to pull to hard, or allowing too much length of vine, that any breaking will cause snap-back. If I think I’m pulling a long runner vine, then both hands will be gloved, and as I pull with one hand, I gather with the other, to prevent any real length of vine swaying around and contaminating clothes.