But what I don't understand is that if you hover over the catalyst on the triumphs tab, it says "Found by defeating the foes of humanity, wherever they lurk." And some have made videos and articles about it, stating you just kill stuff with swords. Perhaps I am getting impatient as my RnG is abysmal.

Looks like you can get it. I forgot that they updated the requirements for the drop/catalyst with Beyond Light. Says that you have to get 300 kills with the sword. Are you making any progress on your catalyst?

Unfortunately I don't have the catalyst in the first place :( My RnG is doo doo.

For me, I've notice that the catalysts drop after the first few clears of the strike playlist for the week on each character.