Are Westerosi armies light on archers in the field, for the time period? It seems like they mention archers and crossbowmen hanging around camps & on walls, but not much about their contributions in battle. Or is that just a function of noble PoVs ignoring their existence & effect, because they don’t like to think about it? W/Could a lord establish a long-term military advantage by encouraging archery in his domain with prizes for village competitions & so forth? Would it be worth trying?

I would not agree with this characterization. We see archers playing a rather significant role in the Battle of the Green Fork, the Battle of the Fords, Redgrass Field, etc. 

Encouraging archery training, or indeed mandating archery training, isn’t a bad idea (it certainly worked well for Edward III), but the only way you’re going to get a long-term military advantage is if you use those archers to best effect (fighting defensive battles, equipping your archers with stakes, combined arms formation with pikemen, etc.) AND your enemy doesn’t adapt to match you. After all, Agincourt and Orleans were only 14 years apart.