Hmm...perhaps your glyphosate solution was too strong and burnt off the tops before it got translocated to the roots. AFAIK bramble roots don't spread by suckers in the way you describe. New growth comes from the central point where the stem emerges from the soil, but I am open to correction on that.
Our garden is surrounded by fields, whose hedges are full of brambles which are a plentiful source of blackberries in late summer, both for us and the birds. Bird droppings at around that time are always purple and full of blackberry seeds, which germinate freely all over my garden :-( . As a result, I'm for ever pulling out seedling brambles and bigger. I wonder if the brambles that you think come from roots of plants that haven't been fully killed off by glyphosate aren't in fact simply seedlings from similar bird droppings.