Emotions are not a nonrenewable resource like oil.
They aren't even a nonrenewable resource in DC, either, because the Blue Lantern Corps rings are perpetual motion engines because the blue lanterns inspire hope in others, which means they make more hope, making more hope is kind of the entire point of the BLC, thematically, and how their rings work backs that up.
Of course, DC contradicting itself is nothing new....
And while I didn't read that storyline, I understand that Relic got a big "I told you so" moment because the DC universe was also running out of emotions, and was only saved when the emotional embodiments sacrificed themselves to restore the balance, so no, it wasn't a "Relic's universe was different" thing, and somehow the fact that the various Lantern Corps should not be using as much emotion as entire civilizations that power everything from their toothbrushes to their air conditioning on emotional energy was also not a factor. So sadly you're giving DC too much credit there.