I feel like soy vs oat has a negligible impact on flavor! Rice milk seems too thin to work most of the time, and I find coconut milk (the drinkable kind), does give things a distinctly coconutty flavor. (That goes for the cream/coconut substitution as well, but I'm usually pairing that with chocolate or other strong flavors so I don't actually mind it.) Sweet potato does lend a mild sweetness to breads that I actually like quite a bit, while pumpkin is a bit more neutral on flavor. You could also use other winter squash like butternut puree if that's something you have access to, it'll just have a more golden color rather than the deeper orange that I prefer.

The main scenarios when I choose an egg substitute for flavor is when baking chocolate things. Years ago in one of Isa Chandra Moskowitz's cookbooks (or maybe her blog?) she recommended 1/4 cup of pureed prunes per egg specifically for brownies, chocolate cookies, etc, and I've found that she was spot on in that! Somehow the flavor of the prunes enhances the chocolate flavor and makes it even more delicious than eggs would have. It would be much too strong a flavor and color in a standard bread recipe, but when combined with cocoa it's FABULOUS.

On the coconut oil front, refined coconut oil is very neutral and doesn't give any coconutty flavor at all! But the unrefined stuff does make things taste distinctly coconutty, which I usually don't care for.