Why is this warning displayed ? What's the purpose?
The message says exactly what it means: a seat in the requested class may not available on one or more legs of the flight. The cabin may be sold out, or more likely, a cabin is not available on one or more segments because the aircraft assigned does not have it.
When you search for flights, you are arguably searching for fares, rather than a particular seat. You can buy a business class fare, but if the aircraft you are on does not have a business class cabin, then you will receive alternative accommodations, and provided it is the regularly scheduled aircraft, you would not receive any compensation for it.
After all, like its competitors, American Airlines offers multiple types of business class on different routes (Flagship Business International, Flagship Business Transcontinental, and short-haul international Business), and does not guarantee you will get any particular product even when there is a business class cabin. More generally, what a business class (or premium economy or business-first or whatever) product consists of in general is a matter of marketing, rather than any kind of regulated standard. On the domestic segments, you would most likely be seated in the domestic first class cabin.
For flights within North America, as with flights within Europe, it's relatively rare to find airliners with three or more classes of service; most people (or more to the point, their employers) will not pay for premium service on shorter flights. In Europe you generally have economy and intra-Europe business class. In North America you have economy and domestic first class, as is the case with AA's 737-800 (seat map 1, seat map 2) and A319 (seat map 1), and two of the three A321 layouts (seat map 1, seat map 2). I do not know if any aircraft using the third A321 layout are deployed on South American routes.
How can one tell (without having to check seats) whether business is available or not?
You can't, unfortunately. You need to check the scheduled aircraft type against what is reported at places like SeatGuru. You can search by aircraft type/model, but even then, you can't always tell which cabin configuration the aircraft will have, especially as cabin upgrades are rolled out as aircraft are cycled in and out of service. And of course, there can always be last-minute substitutions.
This is why there have been so many FlyerTalk threads asking about specific tail numbers and the routes where they are deployed and chancing about whether someone will get the newest seat or pod or entertainment system. (For what it's worth, there isn't a reliable way to search by aircraft registration either.)
Does this warning give AA the right to downgrade me later if they feel like it ?
It depends on what you mean by "downgrade." If an international business class cabin is not present on the aircraft, but a domestic first class cabin is, you'd likely be booked into domestic first. If you are on a regional jet that doesn't have any premium cabins (I think only American Eagle Embraer-145s are still configured as all-coach), then you would be assigned a seat in economy. This is not a downgrade in technical terms, though it may feel like it.
I remember scoring an upgrade to business class on a United LAX-IAD flight a few years back. It was the Sunday after Thanksgiving, however, and presumably to increase capacity, they swapped out the 3-class 767-300 for a larger 2-class 777-200, and I was seated in crummy old domestic first, with the domestic first seat and domestic first meal. First World problems and all that. Still, it was a good lesson about managing expectations.