I already posted an actual play example from 4e D&D. The players in my 4e game had their PCs call on the powers of the gods, or draw on their relationships to the gods, all the time. It didn't cause any problems.But effectively you're trying to craft some sort of freeform "bargain with gods" system on top of that. It is something that would be perfectly fine in some other sort of game, one that already wouldn't have a codified system for representing these things. I don't think it is a good idea for D&D.
I mean, from the point of view of mechanical balance, a check on (say) Religion is no more or less powerful than a check on (say) Arcana, which is in turn no more or less powerful than a check on (say) Athletics. The differences pertain to the fiction. And those differences included differences of consequence. It's to be expected that trying to perform a feat of heroic strength, and failing, is likely to have a different result from trying to invoke the power of the Raven Queen, and having her ignore or deny your plea.
To me, you seem to be describing an approach pretty much along the lines of my 4e experience.my idea didn’t involve Odin simply granting the request. As I said, I’d have involved a cost to the character.
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I think for clerics and warlocks and other characters who are associated by a relationship with an otherworldly entity, that relationship should be meaningful. So when there are opportunities to being that into play, I’m usually pretty happy.
Maybe I'm misreading, but I get the impression that you would be quite happy to live with more proactive players.If my take on this prompted players to try and leverage their characters and existing related lore to try and solve problems, making them more proactive as a result… well I’d just have to learn to live with that!