Welcome to outdoors.SE!
It occurred to me that the entire weight of a climber is resting on the strength of the ATC clip and not on any carabiner.
This is not quite right. My first quibble is that in the situation you describe where there was slack in the rope, the load is dynamic, so it's much greater than the weight of the climber. (Sorry, I'm a physics teacher, so I get pedantic sometimes.) The second issue is that it's not true that there is no load on any carabiner. The carabiner that is clipped into the ATC is acted on by a downward force L equal to the load of the falling climber, an upward force A from the anchor, and a small force B (probably upward) from the brake strand. The ATC is designed to make the rope jam while B is still very small. So to a pretty good approximation, Newton's first law applied to the carabiner says that A=L, and the carabiner feels a stress L, i.e., it's fully loaded with the dynamic force of the falling climber.
I think it's true that in this situation, where the second falls and the belayer has left slack, the force transmitted to the anchor can be considerably greater than the force that would be transmitted to the belay station in a normal lead fall. To clarify this, let's consider a normal lead fall.
In a normal lead fall, the rope forms an inverted U around the top piece of protection, and in this U configuration, the friction in the U causes the tension in the belayer's strand to be lower than the tension in the climber's strand by a large factor. The exact factor depends on coefficients of friction, but sources I've seen for typical friction between a steel biner and a climbing rope put it at about 5 to 13. Although this is nice for the belayer because it greatly reduces the tension in the rope at the belay station, it comes at a price, because force is bring transmitted to the top piece of protection, which could pull out.
So in your situation with the second falling on a slack rope, the good news is that he's not loading some crappy, non-redundant nut or cam that was placed somewhere because there was no opportunity to do any better. He's loading the anchor at the top of the pitch, which is presumably redundant and carefully constructed.
I guess you were concerned about the stresses in the block of the ATC itself, and whether they would crack the block open. The thing is, these are forces exerted on the ATC by the rope, and by Newton's third law the ATC is exerting equal forces back on the corresponding parts of the rope. The block of the ATC is made of metal, and the rope is made of nylon. If something is going to fail in this part of the system, I think it's the rope, not the block of the ATC.