I agree with Sheldon's assessment and with that of John Allen on this point. The main issue with AW hubs jumping into freewheel is mal-adjustment of the cable leading the bicycle to slip into neutral from normal (2d) rather than high (3d). Most riders of old 3-speeds are hobbyists who are just riding around for fun and not exerting the kinds of forces needed to thrash the clutch into neutral from H.
When you have a hub that is slipping, you clean, lube and adjust first. Blow any old grease or gunk out of the hub with an inundation of WD-40 or similar. Provide fresh, 20-weight oil in the port, and then double check your cone and shifter cable adjustments. (A more advanced version of this involves greasing the newly cleaned outer cones and bearings, but that's an aside). If the hub slips from H to neutral, the cable is too tight usually; if from N to neutral it's too loose.
The AW's "neutral" comes as the product of cost savings in production. When you build a hub like the old AW, you need some way to prevent the simultaneous engagement of two gears at once (which would damage the hub). The AW forces the hub into one gear or the other when you are "in between L and N".
But when you go to solve the "two gears at once" problem between N and H, you need to decide how you want to do that. How much do you want to spend on assembly and production of each hub? How tight do tolerances need to be? The model K hub was in this regard superior to the AW in that the K used a ramped clutch/driver system to force the hub into one gear at a time. But this meant tighter tolerances and assemblies because it's a more complicated system.
The AW solves the issue the cheaper way - leave a gap between engagement of H and N. You can't engage both at once because on the AW, if you are right in between, you end up spinning the clutch in a gap in the hub. This means loose tolerances and cheaper assemblies in production. But the byproduct is you get the false neutral of the AW. The modern "no in-between gear" AW solves this issue by forcing the hub into gear again, but I regard the NIG hub as a later hub and different from the historical AW.
This all assumes you don't have a badly damaged clutch or other engagement surfaces (pinion pins for H gear). If you do a CLA procedure and still have the issue, you may need to tear down the hub and check for engagement.