I plant has come up wild in the yard around here, from pruning it back trying to kill it over the last couple years, I've never really seen it 'bush out' or put on multiple tips. When cut at the base it seems to always just regrow a single vine from the internode below the cut, or erupt another vine from a different location depending on the size of the tuber below the surface (which are generally quite large and wild). There's also some growing in a boundary hedge which I cut more towards head height because they start to lean and try to grow across trees near the hedge. Those only seem to ever respond the same way, with a single vine from the internode below the cut. I don't exactly prune them often, so perhaps a more frequent pruning regime might destabilize their hormone balance and get them to put on more than one tip, but I've not seen this behavior. That's the concern domestication of plants though, they don't always work. Who knows though, you may find that one mutated strain that can't check it's hormones as effectively and it sprouts all over. Just have to experiment like our ancestors did.
Though, one thing you could also try to induce excess budding by forcing the vine to grow more horizontally than vertically. i.e. along a clothes line. Let it grow vertically out of your pot for a few feet, then continually tie the tip down to a horizontal wire for several feet then snip it. This should really throw it's hormone control into confusion and just might spark excess budding. Then it's just the matter of pruning the tips back to the main 'cordon' every couple weeks. These are a weird plant. My neighbor cleared an area of brush several years ago and greenbriars were some of the first things to bounce back, and they could detect where other trees are. Several weeks after his clearing, I was walking by and looked at the area and seen all the greenbriar vines growing towards a Hickory 20' away which was then the tallest, closest object they could detect. It was a crazy sight.
