I received an answer from someone at American Airlines that for Basic Economy fares the seating preference on the AAdvantage profile page has no effect on what seat you will be automatically assigned. I don't know if that answer is definitive, but it seems likely to be correct.
I called the main booking number for American Airlines, the person that I talked to was quite pleasant but didn't know how automatic seat assignment worked and she transferred me to customer service. They didn't know either and they transferred me to web services. That person didn't know either but he was willing to spend quite a bit of time trying to find out, including checking with other people there. After disappearing a couple of times to find out, he eventually came back and said that the seating preference in the AAdvantage profile has no effect for Basic Economy, and that seats for that fare are assigned randomly.
I thanked him for checking, and that I understood the policy, but I did point out to him that it does not indicate this anywhere on the profile screen, or in the details for Basic Economy. I also pointed out that I don’t think the seat assignment is completely random, otherwise in my admittedly small data pool of approximately a dozen flights I would have expected to have been assigned a window seat roughly four times, instead it was zero times, regardless of whether or not the flight was full. He said he didn't know how the seats are assigned. I said that's okay and I'm sure they have their methods, but just to confirm, you are sure that the seat preference that is set in the AAdvantage profile has no effect for Basic Economy, and he said yes.
I have no way to know who he talked to and how authoritative his answer was, but it certainly seems plausible, and it is consistent with what I have experienced. And as pointed out in a comment by @PeterM it makes no financial sense for an airline to automatically assign someone's seat based on their preference, while simultaneously trying to encourage them to pay to select a seat.
That being said, I also don't think that they purposely put people in terrible seats in order to encourage them to upgrade. There is sort of a tightrope walk in marketing where you try and maximize profit while giving the appearance of maximizing customer satisfaction. What I suspect is happening, based on my limited data set, is that for Basic Economy the system tries to put you in what most people would think of as an okay seat, which is an aisle seat as far forward as possible within that price range. If an aisle seat is not available then you will be in a middle seat.
But not a window seat, because apparently from what I have read a slight majority of flyers prefer window to aisle, so my guess is that they don’t automatically assign a window seat in order to keep those seats available for potential upgrade. I guess if you got lucky there would be a lot of people traveling together and filling up middle seats, and you could wind up being assigned to a window seat. A larger data pool might provide some information on how often that happens.
American's Basic Economy fare can at times be quite a good deal when it is available, as long as you accept forfeiting what you spend on the ticket if you have to change your flight to a different day. But it doesn't necessarily mean that you are doomed to a cattle car experience since you can still select a seat for a relatively minimal charge ($18 for my recent flight).
However there are over 100 million AAdvantage members and I suspect that a sizable percentage of them don't realize that the seat type preference on the profile screen has no effect for Basic Economy. My guess is that many of them wait until the 24 hour check in period to find out if they hopefully got their preference, instead of selecting a seat at the time of booking. But by then the choices will likely be very limited at check in time. Now that I know how it works it's a lot more obvious that if you prefer a window seat you will need to pay for it anyway so you might as well select a seat at the time of booking. I would think making this process more clear on their website might even increase the number of people who select their Basic Economy seat at the time of booking.