Please pardon my naivete, but why do I need a third party service to check on upgrades for me? Is this something mere mortals don't have access to, or is it a knowledge base issue?
You don't need a third-party service if you're willing to call AA endlessly.
But it's simply that AA doesn't provide that information itself online. It's considered "geeky" information that "average" mere mortals would not be interested in.
It's in the form of bucket codes and their avaiabilities. Fares are arranged in buckets, correlating (for paid fares) with the first letter of the fare code. AA does show you this for paid fares while booking (on the summary page after you select flights in both directions, assuming it's a round-trip booking). It's typically stuff like Q, N, etc, for discount economy fares, for example.
What you need for upgrades, though, are "hidden" buckets which (because they're not used for sold tickets) are not available to typical travel agencies (and other people who have access to "public" buckets). So you need a special third-party service to see these "hidden" buckets.
Buckets are followed by a number from 0 to 7, which means how many seats are available at that moment in that bucket. (7 means "at least 7"; you can't tell the difference between 7 available and every seat on the plane available.)
The AA "hidden" bucket for upgrades from coach to Business class (the second class up on international flights, as well as on three-class domestic flights, though the latter you won't likely run into when leaving from the middle of the country) is "C". "C0" means no availabliity, "C1" means only one seat available for upgrade, "C7" means at least 7 seats available for upgrade.
The AA bucket for upgrades from coach to first class, on domestic two-class flights, is "A". This bucket happens to also be used for some paid fares, so it's much rarer to have a problem with "A0", while it's very common to run into "C0".
So now, to see upgrade availabliilty on an AA flight, you have to use a service such as ExpertFlyer and enter your origin city, your desitnation city, choose AA as the airline, optionally specify a connection city (if you want to narrow down), and specify which award you want (check the ones corresponding to "C" and "A"). Then you run the search, and you get "C0" or "C2" or whatever, on a per-flight-leg basis.
But if you get "C0", you have to search again, perhaps looking for another routing. Or maybe search again tomorrow.
So while in theory you can do this on a 5-day free trial, in reality 5 days may not be nearly enough. And thus you need to pay.
Meanwhile: Once you find availability, you cannot use it
instantly AFAIK. I think you have to book the ticket and wait for it to go from Purchased to Ticketed before you call in to ask for the upgrade. (At least I know that one time years ago when I didn't do it in that order, it got all messed up. But maybe they've changed it since?)
Because of this delay, if I want upgrades that can be confirmed at booking time, and I only need "C1", I still prefer to see "C2" or "C3" rather than just "C1", because there's always a chance that someone will grab a seat or two between when I run the search and when book, wait for "Ticketed", and then call in to request the upgrade.
If no upgrades are confirmable (all "C0"), you can still ask to be put on a wait-list. But that way, you're not sure if you'll get the upgrade (and your chances go down the lower your elite status at AA, with the lowest chances if you have no elite status at all). Thus if you have no elite status, confirmable upgrades ("C1" or preferably higher) are IMHO all the more important to know before you choose your flight.
(Of course, I'm not sure how much flexibility on dates you have. But you presumably have at least some flexibility on routing. But remember, only flights operated by AA qualify for upgrades, so you can't book a BA or IB flight number, or an AA codeshare operated by BA or IB, and then expect to be able to upgrade it.)