The Exigence itself is a portion of power the god can tap into to bestow, but not use for itself. When Ten Sheaves empowered Janest, she came out of the deal with more power than Ten Sheaves himself had.
1) Gods destroying themselves in the creation of Exigents is not normative. Most gods create Exigents and come out a bit weaker, and go "Yeah, not doing that again."
2) Gods don't want to permanently weaken themselves, so Exigents really only happen when a god needs a champion right now or when a god has put a lot of thought into the matter and decided the long-term benefits of having a champion outweigh the costs.
3) Divine psychology is influenced to greater or lesser degree by divine nature and portfolio, varying from god to god. Plentimon is the sort of god of gambling who's going to bring gambling into stuff even when it's maybe not a good idea for him personally, because he likes to gamble for gambling's sake. This is not reflective of all gods.