First a disclaimer: I KNOW that viewing the sun through a telescope will burn my eyes, so no need to convince me of that. I'm not going to try it. This question is about the optics and physics of why that is true.

A friend and I were discussing ways to safely view the upcoming solar eclipse, and I discovered a huge gap in my understanding of optics theory. Namely, these two truths:

1. Telescopes cannot increase the surface brightness of an image, they can only increase total brightness by making the image bigger at the same surface brightness. No point in the bigger image will be brighter than it is with the naked eye, due to the conservation of étendue.
2. If you look directly at the sun through a magnifying telescope, you will quickly burn and permanently damage your eyes. However you can get away with looking at the sun with the naked eye for a second or two and not immediately go blind.

I accept both of these as true and am not challenging them, but seem to me to be in contradiction to each other, even from a purely optical and physical standpoint (the biology of the eye and human visual system being irrelevant). What am I missing?

To further clarify my question / confusion, consider this thought experiment. Let's construct a camera out of a closed cardboard box, with a small hole + lens on the front, and a simple sheet of paper on the back (inside) acting as the "film". Scenario 1: We point this camera at the sun, and it makes a tiny bright circle on the paper (a tiny image of the sun), but not bright enough to ignite and burn the paper because the aperture and lens are too small. Scenario 2: Next, we place a telescope in front of our camera, which magnifies the image and creates a larger circle on the paper.

The key question: Will any point of the paper in Scenario 2 become hotter than it was in Scenario 1? Is it possible cause the paper to ignite?

Truth 1 would indicate that the answer is no, while truth 2 would indicate that the answer is yes. What am I missing?