Alton Brown suggests adding aromatics to the turkey include sage, apple, cinnamon and onion. If you put these ingredients into the turkey’s cavity, how exactly does that flavor the meat? I’ve always wondered about this…
I’m not sure that it does in a way that would be actually discernable in a double-blind testing - but the effect of the cooking aroma produced that wafts throughout the house is not to be discounted.
From my own experience, it doesn’t, at least not to any degree that I’d notice. I think it’s just a way of pushing more products for your local supermarket.
I tried some of Jamie Oliver’s gee-whiz-wowie Christmas recipes last year. A lot of extra effort and expense for very minimal return on my investment. The way I’ve always cooked Thanksgiving and Christmas dinner is just fine, thank you!
The apple and onion contribute moisture to the inside of the bird while it cooks, but only a little flavor. I use this recipe and I’ve always loved the results. Last year I accidentally ruined my turkey stock because I forgot about the cinnamon stick. Whoops! Anyway the flavor of the bird is good, but you can use lemon or orange instead of apple and rosemary with the sage instead of the cinnamon stick.
Just as the name suggest, aromatics contribute mostly odor, thanks to the essential oils in the onion, herbs and spices (and a little bit the apple, although apples aren’t brimming over with essential oils) being released into the moist heat of the oven and odor is mixed up with flavor in our brains. It’s not overwhelming, because it’s not meant to be - if you want cinnamon flavored turkey, you’ll have to use powdered cinnamon and rub it on the turkey…and I won’t be coming to dinner. 
Think of it as perfume for your bird.
I’ve made Jamie Oliver’s roast chicken before, where you stuff the body cavity with quartered lemons and rosemary, and you can definitely taste it in the meat.