Humanity is Great, but Humans are a Pain

Question: Not every god has Scions, right? But what keeps them from having a favored mortal or special pet on earth that they care about, or is running the Titan war too costly for such pleasures?

Nothing is stopping them, but it's probably pretty rare, for a combination of reasons including time, energy, interest and investment.

For most gods, the amount of time and energy they'd need to invest in a purely mortal companion or pet is just not worth it. Going to the World is a pain for a god, who has to pay resources to get down there and who knows that, thanks to Fateful Aura, the odds are always that someone or something is going to jump him during his vacation. Gods that don't have the Avatar Birthright almost can't do that, since their overwhelming power level would blow up society around them and they'd be at the mercy of vicious Fatebinding from all sides; gods that do have Avatar have to spend for it, and while it helps reduce the issues of being divine in the World, it doesn't make them go away, so she's still looking at dodging Titanspawn attacks or major world events while having to watch every step to avoid accidentally Fatebinding herself into something that she'll regret. These are all the reasons gods have Scions in the first place, so they don't have to come to the World and deal with all its bullshit; coming down for literally no benefit other than a brief vacation with someone they like is, for most, going to be more stressful than anything else.

You're right, also, that the fact that the gods are currently embroiled in a massive war with the Titans means that they just don't have a lot of energy to spare. Gods who are blowing Legend and Willpower to shrink down, go hang out in the World and avoid rampaging around from their Virtues are gods who are not going to have those resources available when something big goes down in the Overworlds, and while they may not spend too much, if their enemies haven't been similarly frittering away their resources, the gods are going to come up at a disadvantage. Then, too, there's the issue of being in the World on holiday instead of on the front lines of the Overworld, potentially meaning that a Titan's assault might break further through or that a goddess won't be on hand to help save the day if a massive disaster occurs. Sure, she can run back to the Overworld as soon as she finds out - but precious minutes will be lost while she does so. Maybe she'll get there in time and everything will be fine, but maybe the Titan will gain ground and destroy more of the Overworld than it would have if she'd been there, or steal some important item she could have been helping to guard, or even kill another deity that didn't have her for backup. Going off to hang out as a Legend 4 in the World when Cronus might come rampaging through the front yard at any moment is a risk many gods simply aren't willing to take.

And that's assuming that this god who wants to hang out in the World isn't even doing anything else, which is almost never the case. There's a war on, there are Scions rising to power everywhere, and the global politics of the divine world have never been more active; there are precious few gods sitting around in their Sanctums wondering what to do today. They are busy as shit, which means most of them don't have time to go maintain some favorite mortal in the world.

And speaking of mortals, that's another issue: most gods simply aren't interested in maintaining a favorite mortal, and that's because mortals are boring. It's not their fault; they can't help it. Gods have Epic Attributes at levels that mortals can't even comprehend, much less deal with; a mortal can never have an interesting or stimulating conversation with a god that has three or more Epic Intelligence, for example, can't physically keep up with a god with even basic Epic Stamina and probably moves at the speed of an epileptic snail both physically and mentally when compared to divine beings, even if they're not very good at those things themselves. It's very hard for gods to have meaningful relationships with mortals; they can have empathy for them and care about them in broad strokes, but it's hard to really connect with someone who has the brains of an amoeba compared to you, or who's so fragile that you might accidentally sneeze and blow him up.

This doesn't mean that gods don't have mortal friends and lovers, of course; they often do. But they don't maintain them over long periods of time; their lovers generally tend to be abandoned as soon as they get them knocked up, or if they're really attached to someone they usually just drag them off to become Legendary in some way rather than deal with the annoyance of their mortality, thus removing them from the World entirely. Most often, the mortals simply die - not of old age, but because hanging out with gods is very hazardous to a mortal's health.

Which brings us down to the fact that most mortals can't survive being a god's favorite pet or lover for very long. That god's influence and Fate drag them in; they become tragic victims of it, whether because the inevitable Fatebound role they fall into demands it of them or just because the god's Fateful Aura always brings trouble and danger down on them just for being too close to him. Of course, the god could decide to really safeguard and defend this mortal from all comers, especially if he has Guardian or is willing to just be there in person to deal with it - but that goes back to the other problems of maintaining mortal pets, in that he's spending a bunch of resources, risking damaging Fatebonds and neglecting his other duties, goals and the war he's supposed to be fighting to do so. For a god, having a mortal pet is like owning a Faberge egg balanced on a three-foot-tall pin in the middle of a giant room full of cats; he almost can't do anything else but hover over it if he wants it to stay intact, and even then the odds are not with him.

What you'll most often see gods do in myth is remove the mortal from the World and try to turn them into something more interesting and durable. Most often, this involves making them immortal - an easy enough task for gods with Health and/or Magic - and putting them in a Terra Incognita where they'll be safer and easier to visit. Even this isn't common, though, for all the same reasons above - they have to spend resources and take time to go do it, so they probably won't bother unless this particular mortal is incredibly important to them. And even that doesn't always end well, as the myth of Eos and Tethonis attests.

You can, of course, totally use the idea of a god's mortal pet, companion or lover in a game; just because it doesn't happen often doesn't mean it doesn't happen ever, and for purposes of your plot you might happen to be involved in a situation during the brief window of time that a god tends to hang out with a favored mortal before moving on. Gods are out making Scions for this war, after all, so some of them probably have a current mortal lover that they have an interest in seeing survive to bear children, at least; others might be safeguarding untapped Scions, mortal for all intents and purposes, until such time they are needed. And once in a while you might have a god who just genuinely really loves a mortal and wants to hang onto him or her, and build your plot from there.

But these things are probably pretty rare. For most gods, mortals are useful for creating Scions and occasionally entertaining, but they just aren't worth the prolonged effort, irritation and resources to keep holding onto a particular one instead of working on shoring up their power and keeping the entire World intact by focusing on the war with the Titans.