Well, imagine this:
When the gods sought to overthrow the Primordials, they used the mightiest tool of all times - one that allowed them to channel their very being into mortals, the only way they had to bypass the geas laid upon them. This tool was the Exigence, and after the war it remained in the custody of the Unconquered Sun.
You can have the Exigence as the greatest creation of Autochthon. It's easily the perfect tool because it's capable of shaping essence like nothing else, it can instill creativity and life and all those other things. It's even a perfect contrast to his sickness and fits very well with his theme of surpassing his limitations.
Or you can leave that out, the origin doesn't matter as much after all. Maybe Ignis Divine created it on his own, maybe the other Incarnae helped, maybe it was even a collaborative effort by all the gods or came from somewhere else entirely.
Then this mightiest of gods actually allowed lesser gods to wield this power. This actually fits really well with the role of the Unconquered Sun as some good, virtuous person.
And it puts a wedge into "Solars are perfect for everything". Yes, even if Solars are better than Exigents this holds true because even then, UCS recognized the need for more.
Plus, you can put all sort of power politics into it if you want to be more nefarious. From "leave me alone to my games, here's a tool for that" over "I did it to undermine the other Incarnae" and a whole bunch of other things, all that fits with Ignis Divine granting other gods access to the Exigence.
So what do we actually get out of Exigents? Both for the setting and individual stories.
As just mentioned, they're simultaneously tied to Solars and sidestep them. They are also a type of hero that's new to the Age of Sorrows, so it's a counter to "everything revolves around Solars" and "Solars are the only heroes that can save us".
It's already been hinted at that getting an Exigence can be as much a matter of celestial politics as anything else. Getting celestial politics more involved in creations story is a good thing in my book, as long as it's not overdone and turns celestial politics into all that matters. Well, the Exigents actually accomplish that.
It's a challenge to players independent from Dragonbloods - which are too tied to their faction - Lunars (same in some ways) and Sidereals (which don't rule territory in Creation). At the same time, they're more human than gods are and can tell different stories.
It actually allows small gods to have an impact on the setting beyond just getting curbed by every major thread. The story of Janest is a great example of that.
And generally, we just get an entire new set of supernatural actors to play with.
As for actually playing them:
I imagine they'd be great for lower-scape stories than Solars or Lunars.
You can't really play a Solar as "personally empowered by a god". As in, the character probably didn't know the Unconquered Sun on a personal level, Ignis Divine is obviously not a local god, Solars don't really get sent on personal missions like that. Neither do Lunars. Sidereals are fated to exalt and mostly tied to Yu-Shan, while Dragonbloods are too tied to their lineage, even if you play an outcaste.
Again, look at the story of Janest. Sure, you can also tell a story about a farm maiden who one day exalted as a Solar to defend her valley against the Raksha, and who then did that. But it'd lack the personal touch we see in the story, and with Solars you'd break that scope quickly due to various reasons. Your power would exceed that relatively small ambition quickly - and you have that giant wyld hunt target painted on you.
So with Exigents, you can tell different types of stories.
An Exigent can be the chosen companion of a local god, who the people there mostly know as that young man who lives secluded in the woods, is wiser than any village elder, and who often comes into the village to aid them while defending them from attacks. Sure, you can kinda-sorta do that with a god as well, but a god would be less tied to their affairs and the issue of worship would come up. And you'd not have a "chosen mortal".
And this would also work less well with a god-blood, since you'd automatically have parent/child relationship there.
An Exigent can also rebel against their god in a much more direct fashion. You can't tell the story of a starting Solar who actively fought against the very god who chose them. You can do that with an Exigent because that god might be much less powerful. Since it's still an Exaltation and not actually dependent on the continued gods existence, this works very well if you want it to. Sure, a Solar or Lunar can overthrow gods as well, but that's much less personal and wouldn't have the sweet result of "overthrow the one who made you what you are".
Again, you could kinda-sorta do this with god-bloods, but since being a god-blood isn't really a matter of being chosen (you're born that way), it's not the same.
And an Exigent could potentially have directly earned their Exaltation through trickery, or in a contest, or by some other personal means.
You could have a person who managed to out-play a god with the specific price of Exaltation in mind. That doesn't really work for Celestial Exaltations since you can't challenge the Incarnae, and while you could get a Solar Exaltation for managing to best a god, that's not the same since it's then still granted to you by someone else.
There's lots of interesting options for play - both for PCs and NPCs and I think Exigence also does some interesting things to the setting.
Oh and thanks to
@Fenrir555 for the encouragement to actually write this post.