Playing as a primitive living as far back as 3200 years ago... That's what we're doing. The fact is, our characters probably would be considered heretics back then for our failure to consult the Gods on virtually every decision or action. The game is very liberal in not requiring many secular activities.
It is anecdotal, of course, but my experience leads me to expect a smoother passage through the difficulties of the game solely by attending to the sacrifices 'religiously'. I've received actual good intelligence regarding game whereabouts from spirits that held me in good standing. I'm not sure how the game would differ at all if the spirits were unhappy with my characters.
Regarding the various hunting and trapping enhancements that my npc peers offer me from time to time, in the guise of 'the best way', I could not vouch for any of them. I've never felt that my traps caught more foxes or found that a lynx hunt required a feather. Perhaps it makes more of a difference when those skills are relatively low? At any rate, in the game experience it is usually not possible to quantify the effect of the spirits unless one is actually entertaining the spirit's presence.
The fact is if you hunt/trap the animals you see, where you saw them, as in from a distance on the strategic map, the odds favor your success. Otoh, if you simply drop traps randomly, or just tramp the tactical map looking for tracks to follow, the hunting/trapping is probably going to consume time and materials and deliver less success. However, if a deer, elk, or rabbit happen to wander near you after you appease them with offerings or attendance, who's to say it wasn't the spirits favoring you?
For that matter, what does constitute the anger of the spirits? Sometimes the forest seems frightening, sometimes I'm awakened by strange noises. Are the spirits roused against me? Because I never chop my own foot with an axe or fall through deep ice as a consequence. It's possible that the 'activity' of spirits may be nothing at all in effect, just a factor in role playing your character.