Yes, but as you said, Odin doesn't care. If you died in battle he'd still come for you. So why not say that you only worship Jesus to hedge your bets?

If only one of the religions requires full commitment, then ironically that makes it easier for that religion to spread. The extremist intolerance and gatekeeping of heaven lets them attract the weaker people in society because there's no Valhalla waiting for them regardless, and they are easy to turn into followers too, because they're used to it already. Like sheep, they just need to be herded in any direction.