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Can someone explain why the second puzzle would need this fancy three-dimensional solution? The area of each strip doesn’t seem important to solving it:

We need to cover the circle as efficiently as possible. That means having exactly one layer of strips. Zero layers doesn’t cover it, and two or more layers are wasted. As soon as you start using strips at different orientations you can’t escape an overlap somewhere. So, clearly the optimal way to do it is to use some number of parallel non-overlapping strips, and their total width will be the diameter of the circle.

Not sure if this isn’t rigorous enough or something, but it seems perfectly clear to me.

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It is absolutely not rigorous enough, I'm afraid.

Sure, it feels wasteful to cover any part of the disc twice. But it also feels wasteful to cover bits near the edge with strips that have only a short length of overlap with the circle. It's not obvious that there isn't some clever way to reduce the second kind of waste that requires you to commit the first kind.

Perhaps the following observation will help. Replace the circle with a "+" shape made out of five equal squares (like the cross on the Swiss flag). The most efficient way to cover this with strips is (I think -- I haven't actually tried to prove it) to use two strips "along" the arms of the cross. Those overlap in the middle, but they still do the job more efficiently than using (say) a single strip of 3x the width.

So, how do you know that nothing like that happens with a disc instead of a cross?

That’s a great explanation, thank you!

I feel like it should be possible to pinpoint why this can’t happen with a circle. Specifically, it seems to require some kind of protrusion in the shape, that ends up being more efficient to cover in a direction that creates an overlap. But it’s clearly not as straightforward as I thought.

Any kind of "pinpointing" you can do is probably going to end up equivalent to the project-to-a-sphere method, except less clear or pretty.

I'm not so sure about that. I don't think introducing an extra dimension is elegant. There are probably more elegant proofs.