It doesn't take hot water to bring on illness or fatigue, although hot tubs are definitely a no-no. No one's said anything in this thread about overheating. Most kids who upchuck after swimming are dehydrated and have excess lactic acid in their bodies. Which is why you should always let a young swimmer use the restroom when they ask - that's how they can eliminate the problem and save the pool staff the trouble of cleaning up yuck. (I was a swimmer for ten years and a age-group/head coach for 12.)
There are valid reasons to avoid combining swimming and skating within a short time frame. Most coaches who travel to competitions with students will put restrictions on their swimming at the hotel. With synchro, I think the restrictions are also to prevent injuries from fooling around - put 16 teens in a pool and someone's going to get pushed or dunked too hard.
Lactic acid will build in muscles as they warm up and expand when you swim, even in the 70-degree pools. That's what causes that weary feeling after exercise. It's great for stretching muscles, not so great for contracting them for sharp movements.
Swimming also uses muscles differently than skating. Think about the flutter kick vs. toe-assisted jump. They are completely different - one points your foot constantly, the other blunts, then points, then blunts again.
Trying to go from one sport to the other too quickly, without proper stretching, hydration and cool-down/warm-up will make your skating less effective. Swimming is awesome cross-training for skating, when done properly as an off-ice workout. Many dryland training drills correspond to skating off-ice drills, too.
I agree that you don't want to do strenous exercise before skating, but it's far from ridiculous, as you called it.