FromScratch, I have never tried baking that cake without wrapping it, so I have no idea if the quality suffers without that step. The reviewers at egullet, who originally tweaked the recipe, stated that the step (and subsequent freezing) was crucial, but that's about all I have to go on. I know there is much debate about wrapping cakes but I figure a moist cake is a moist cake, whether it's baked that way or results from trapping in its own moisture, so I don't personally feel it's a high-risk venture; the moistness itself is the risk, not the trapping. Moist things do spoil faster so if you're worried about it...then refrigerate it the way you would a cheesecake or anything else more perishable than standard cake+cookies.
For what it's worth I do definitely get skeeved out by other things though, like raw eggs (I always use pasteurized in IMBC) and leaving out perishable fillings, so I understand your concern. I'd say, make a 1/3 batch (since you won't have to split an egg) and give it a try; at least if it isn't as delectably moist as it turns out for the rest of us, you won't have wasted too many ingredients and plus I'm sure it would still taste great with ice cream, etc!