@npub1nhn6...wgwl from my understanding of how glyphosate works... if you paint the leaf with glyphosate, everything connected to it will eventually die down to the roots.
The only other cure for convolvulus I'm familiar with is planting a honeysuckle next to it in the hope that it will entrap the convulvulus into a doomed romance leading to both of them pulling themselves out by the roots (Flanders, M and Swann, D; 1956), and that may not be practical in NZ because I have a vague recollection honeysuckle has the potential to be an invasive weed itself.