I haven't thought about leaving it uncapped overnight to let it areate and check out how it affects the taste. I should know better from having actually worked distilling water and testing it onboard a ship many many years ago.
There is such a thing called a deareator that takes out some of the oxygen from the water. Oxygen is a key component for oxidation (rust) to occur so taking it out slows down oxidation in plumbing systems and machinery that uses water as a cooling fluid, but it sure makes the water taste flatter than a pancake.
I currently have access, at work, to very high quality R.O. (reverse osmosis) water that I fill and drink for use at work and at home even though we have commerical bottle water dispensers everywhere. Most everybody that drinks that high quality R.O. water, comments favorably on how good it tastes. They are surprise by it. Not many people pay that much attention to the taste of water up to a point but there is a big differerence in the quality of R.O. systems from what I've been able to tell.
Sorry, I posted and didn't really address your question. I don't know how Aquafina is bottled. I'm guessing it is through R.O. but I don't know. Some bottle water is mineral enhanced, some is natural spring water. Some is artesian. Some is that fancy woo hoo carbonated stuff. Some schisters actually just bottle water straight off the tap and sell it that way.
Filtering water off your tap is the least expensive way to try to clean water. I am talking about the ones you attach to the end of the faucet. The taste of the water will ultimately depend also on the initial quality of the water before running it through the filter, that is, where you live. If you are particular about the Aquifina brand, I doubt you will be happy with a simply tap water filter.