Welcome,
timiny!
I'd guess some call and put AA staff to work, others use the website to do some checking (though IMO it's not very good for that purpose - CO seems better set up, if more restrictive, with theirs,) others use a third party service to check before they call - for $ ExpertFlyer, free KVS tool, etc. (but I am not sure how the latter works or if it is current - a search would reveal that.)
I think effectiveness / success depend on itinerary and date flexibility, people skills (it makes it easier for an AA agent to serve us if we make it easier to do and personalize things positively, IMO,) patience and either long-term planning or being able to be an opportunistic flyer and to move fast... and a shot of realism.
"I am a AA Gold, my dog goes to Gerard Arpey's dog's veterinarian and I need four SAAver award seats for me and my family in the J cabin on ORD-FCO Saturday June 17 to attend a wedding, returning Sunday July 2" being among the least effective.