Plastic hub adapter on aftermarket rims—is it needed?

"... is the ring a load-bearing component, thus making it unsafe to drive the car without the ring installed?"

tl;dr: Not exactly but close enough. You do need those rings and, if you lose one, you should replace it.

Over at ECS Tuning, they have a post about this exact issue. Quoting from Why You Need Hubcentric Rings for Aftermarket Wheels:

When the hub is smaller than the center bore on the wheel, the wheel will not be technically centered. This causes vibration in many cases, especially around 50-60mph, that feels like a wheel imbalance. The case is, in fact, that the wheel is slightly off-center and not making a smooth rotation. With hubcentric rings, you fill that void and adapt the center bore of the wheel to the hub bore of the car.

To oversimplify a bit, while you are correct that lateral loads are handled by the lug bolts, those are just keeping the wheels attached to the car. They don't actually keep the wheel aligned with the axle. This is a consequence of the fact that the wheel has cylindrical bolt holes bored through it rather than having threads that you can screw the bolts through.

Unfortunately, if the hub bore is smaller than the wheel center bore, you have effectively given up the foundation of the wheel-car system. The wheel will have just enough play to move slightly off-center when under loads. If this happens at the right resonant frequency, the wheel will feel like it is imbalanced and, in the wrong circumstances, can make the vehicle difficult to dangerous to drive.

I remember when I saw the centering rings on a wheel set that I bought some years back and I was also skeptical. I wondered first if they'd function in any useful way (yes, they definitely help center the wheel on a slightly smaller hub) and if they'd last (yes, proof by existence). The second point makes sense if you think about it: you have some pretty serious industrial plastic that is largely only presented with compression loads. There aren't a lot of motions that could wear away at that plastic ring.

If you're worried about losing the rings, my suggestion is to call one of the zillions of online retailers (e.g., I got my wheels from TireRack), tell them that you have this car with these specific wheels and could you order a set of centering rings. It's a mainstream part these days, so there's no reason that you can't have a spare set to prevent worry in these anxiety-ridden times!