The growth in 19th century is much more explosive, and I would say that the primary reason is widespread use of coal for energy. Not being dependent on the power of muscles (and wind) was a game changer for Europe. And absence of that capability put a hard limit on Roman economic power. You just don't gain much economic efficiency from burning wood and working people and animals to death.
Economic development in countries that were able to harness steam power was enormous regardless of their slavery laws (some notable industrial hotspots like Prussia never had slavery to begin with). Countries like Portugal and Spain, with their old colonial empires, were much more stagnant.
Generally, slavery only sort-of works in agriculture and possibly mining. Anything more value-added requires voluntary, educated workforce. Nazi and Soviet slave-powered industries were extremely inefficient, sabotage was a constant problem, even though punishments were heavy.