If so, logically he would take
insane damage from what his ordinary, rotating elemental weakness at the time is.
I feel like it's reasonable to view "Ability to gain press turns when striking weakness" as an offensive property of DDS characters, Abbadon example aside. I mean, how does it contrast from Arnaud's Exploit Weakness ability
It's listed as a specific property of Arnaud. i.e. it's unique to him (and Raquel, but clearly 2/5, and also clearly both exist without it for a long time). A bit harder to make that argument of an entire cast (not just PCs but enemies too).
Besides, uh, the Abaddon point does exist, which only weakens your argument. I'm nearly certain Abaddon's not the only such enemy who behaves like this, but I can't produce any concrete examples so just figured I'd point to something with a stat topic reference.
Plus there's my previous point. What qualifies as a "weakness" for a DDS character to hit? Obviously Abaddon doesn't, that's a pretty big problem right there. Does Ricardo, with only 1.25x weakness? Does an OB character with a 1-2% weakness? So on. It's silly, you can't decide. The much more logical thing to do is conclude that giving the opponent a press turn is at least partially the property of the defender.