Yes I do fly for SWA. I think and perhaps TR4A can correct me if I am, as is often the case, wrong. The autothrottles were not originally activated when we started the 737-300 program as the launch customer. The reasons were explained to me that coming from the 737-200 without FMS, LNAV, Autothrottles etc the jump to the -300 would be a big enough challenge without complicating our procedures. The systems differences coupled with new ops procedures would have been a big union issue as well.
I believe USAir was the launch customer for the -300. When we bought the -300 (almost ten years before I was hired) they wanted a common type rating and minimal differences. No glass, no LNAV, no VNAV, no Auto Throttles and no Auto Brakes. The philosophy was "Fly it like a 200". About 6 years ago we got LNAV.