I discussed your experiences with my fiancé and we decided to call SW again to try to get the child fare, but again we got the same rate as the adult fare. The only difference is that it is fully refundable, which won't matter much to them. We'd love to buy her a ticket since they don't want to, but right now that's not really an option for us since our first baby is due in March and with all the expenses associated with babies and their gear, plus paying for our own tickets, my mother's, and the baby's for May and June, we don't really have an extra money right now. We're hoping that telling them your stories about people getting asked for proof of age will convince them that it's not an option, but any other ideas are welcome!
Oh yeah, the kids I have seen who were well over 2 without a seat were not on SW, which could be the explanation. I, unfortunately, was in the row with a 4-5 year old lap baby and on another flight my mom got stuck sitting next to one. So I can empathize with whoever would have to sit in their row, another good reason to get a seat!