To quote wikpedia, emphasis added by me
The main chemical reaction in thermate is the same as in thermite: an aluminothermic reaction between powdered aluminum and a metal oxide. In addition to thermite, thermate also contains sulfur and sometimes barium nitrate
In this thread we use the terminology "Thermite" synonymous with "aluminoferric thermite", and similiarly, Thermate may simply be thermite with sulfur added. There are other types of thermite whose final product is not iron, but can be one of a range of metal oxides that can be used instead of iron oxides, as long as they're used in conjunction with elemental aluminium in an aluminothermic reaction. Thermites without both aluminium and iron may be possible, but also may be uneconomic or may not be able to produce an useful temperature.
The effect of elemental sulfur is to to create an eutectic when cutting through steel, reducing the melting point of structural steel to around 1000 degrees celsius from 1315 when steel liquifies - Needless to say, steel depending on type becomes plastic already a hundred to several hundred degrees below the melting point, a fact that is used in forging.
This following being my own guesswork, corrections and specifications may be posted, the speculative nature however do not whatsoever change the outcome of the final refutation:
The barium nitrate may be added to also cause oxidation of the sulfur to achieve a higher reaction temperature, and possibly increase the rate of the aluminothermic reaction. But to a similiar effect (as an oxidizer), potassium nitrate or potassium permanganate could probably be used as well. Edit: Prof. Jones' analysis actually shows an abundant amount of manganese in an atypical pattern, meaning it's not the manganese that is ordinarily put into steel to make it more machinable.
However, barium is one of the first elements anyone looking for explosives would be searching for, as it's a common compound in explosives, if not in the main charge itself, it'll be found in many priming mechanisms. Furthermore, for the effectivity of thermate and Dr. Jones' findings perfectly corroborate the suggestion of a barium-free thermate.
Ultimately the notion that the noted absence of barium would somehow falsify the notion of thermate being used is just absurd. Why anyone would argue this escapes my mind. Your resistance only makes my penis harder.