Because nearly all passengers who are traveling without proper documentation are doing so accidentally, I’d think. If they catch you at the origin airport (which is, in the vast majority of cases, particularly amongst infrequent travelers who are more likely to err on passport/visa requirement), it’s much less of a hassle in most cases for the passenger to correct the issue, or, if necessary, cancel the trip. So there’s definitely a point to doing the document verification at the origin airport if possible.

Again, in this case, AA could only do a document check, and it would have saved the OP's grandmother a lot of hassle. AA certainly could warn the passenger if her documents don’t seem to be what’s required, but I’m pretty sure that the operating airline will only accept document verification that they’ve done themselves. That is, it’s not ultimately AA's responsibility to verify that the traveler has the proper documents.