In WAR's case, it makes sense because WAR's powers are mostly suped-up natural abilities that people have -- in this case, it's an adrenaline spike, or a second wind (which makes sense given it's literally a suped-up Second Wind). As long as the ability is strictly personal it fits in with natural, non-magical abilities as much as Holmgang as a Last Stand sort of effect or the Inner Beast as an enrage mechanic.
I mean let's be real here, FF14's metaphysics are already a bit bizarre. Thancred, who can't manipulate aether and therefore should not be able to use magic, can somehow turn his life force into invisibility... and we just accept that. By comparison, Equilibrium at least seems probable.
In DRK's case, it feels off because DRK's power, being drawn from dark magic, should always come at a price. They shouldn't have "free" healing; they should only have the ability to rip life force from someone else. Same sentiment why DRK shouldn't be able to heal other people directly, unless that health comes from somewhere else, like vampirism or a personal transfer.
Looking at the original Sole Survivor through that lens, you have a skill that essentially steals someone's soul or last breath in order to infuse yourself with power. The problem is that while it's thematic to DRK, it's also mechanically useless to have a skill that only heals you when someone dies -- hence the consolation effect, despite that free heal not fitting the theme either and the delayed heal being only marginally more useful.
(And now you have RPR who uses that iconography better.)