Though to be fair, it doesn't seem like the DC universe has even tried to reconcile all the difference space mythologies.
Lovecraft just doesn't work in a Supers setting. If it can't be punched or bataranged, its largely boring. But funnily enough I have rarely seen superheroes triumph against 'demonic' threats in the same way they triumph against demigods or such. Upper level demons and devils are something you back into a portal, or cast a spell to banish them or something. You never 'take the fight to them' if you will.
But incongruity is what makes superhero settings kinda fun really! Like what I noticed about the best 'original' superhero settings is they add incongruity into their worlds by design! Its generally the really boring supers that have everything run off one power system.
As for what powers they where mucking with....Tapping into alternate dimensions? The Phantom zone is an example of them tampering with this sort of tech before. Thats also an explanation as to why beings in the phantom zone where also affected: as a dimensional rift it altered some beings in there as well.
Lovecraft works fine in supers, and you described how: You description of how demons are handled is also how Lovecraftians threats are typically presented as well. The heroes can punch them all they want (and they usually do, because it's an excuse for grand apocalyptic fights with massive property destruction where the heroes can really unleash their full powers), but the only real way to win is to drive them away, usually to another dimension.
Wild Talents had a supers classification scheme (contributed by Kenneth Hite), called the "Axes of Design", each of which was associated with a different color. Blue is Hite's term for what you're describing -- how many different and not necessarily compatible super power sources or origins there are. A very blue setting is one with numerous pantheon of gods, magic, alien races, mutations, sapient robots, and a zillion other things like that. An un-blue setting (I forget how he refers to the opposite end of the spectrum) would be one where all powers can be traced to a single origin, like trying to explain all super powers as manifestations of psychic powers, or alien blood. I don't have a strong opinion about which is better, I just think they're different tastes. It can be fun to have a supers setting where everything is justified in terms of a single origin point, and it can be fun to have a kitchen setting where everything is allowed and conflicting things exist in uneasy juxtaposition. Neither is superior.
The Phantom Zone would be an obvious power source, but it would have to be a new twist on how. After all, the PZ is usually presented as a relatively old technology, one that's so stable that it's reliable enough to be trusted with holding the most dangerous of prisoners securely. Some new development, that the Krytopians are using to power something -- though again, we're running into a difficulty, because the Krypton is typically presented as a greatly diminished empire, one that gave up their expansive holdings across space to return home. So why would they need a new and massive energy source? That makes a lot more sense for a growing empire.