Thank you everyone for your responses. A couple of brief comments.
@François Pugh Your suggestion about follow the leader echoes what
@LiquidFeet wrote. I'll definitely try that (either skiing backwards or forward, whatever works better) when my son feels like it. I don't want to complain about conditions over and over again, but it has been very warm recently and the run does not hold up well. My son is less comfortable when it's slushy and/or bumpy, and I don't want to push him out of the comfort zone too much. Once it's colder and grooming holds up better, it will be easier to try new things with him.
@crgildart I totally get the point that ski schools are as much for us as they are for kids, but my work schedule it fairly flexible, so I can go skiing alone on a weekday and ski with my son on a weekend, and I'm less pressed about having time to ski on my own. Also, when I was growing up, there were pretty much no ski schools in Russia, so my sister and I always skied with our parents.
@Mike-AT The problem with the ski school at our resort is that when we tried it, they took kids to a less that 100 feet (or 30 meters) long bunny slope, let them ski there, but didn't really teach anything new. I'm all in for a ski school if it gently teaches kids to ski better, but it just doesn't make a lot of sense to pay quite a lot of money (the price includes equipment rentals, and even though we have our own skis, boots, and helmet, but there are no discounts for us) if it doesn't teach anything new. Sorry if that sounds a bit grumpy
Btw, I've been to Austria three times (though it was long ago), and based on what I've seen, ski schools at Austrian resorts are typically very good.