It turned out that in that version of IE, the console object only existed when the devtools were open. So a stray console.log call was causing an exception because console was undefined. But when I opened the devtools to see what was going on, console existed and so everything ran just fine.
I suppose that's the kind of error that only bites you once. When you know about it, you tend to never make that mistake again. But that first time you run into it can cause a painful few hours of head scratching and attempted debugging.
Funnily enough, I ran into a React Native error like the one described in the article, where something worked only while debugging. My previous experience with the heisenbug in IE led me to immediately dig into what was different in RN with the devtools open vs without them. So in a way, those hours I wasted in IE back in 2010 prevented me from potentially wasting hours debugging RN in 2017.