So there are two questions mixed up in here - the first is about traversing the atmosphere without burning up, and the second about traversing the Van Allen belts.
It’s true that re-entering the atmosphere from space is a delicate business, and there are only a few safe paths to do so. The atmosphere, as easily as we move through it on the surface of the Earth, can pose a significant barrier to fast-moving objects. Air resistance is a major factor in designing everything from cars to parachutes to space shuttles. If you’ve ever been out in high winds, you’ve felt the kind of barrier wind can produce to your own motion, and how much force it takes to move in resistance to it.
Objects which encounter our atmosphere from space are generally travelling much faster than any winds we’d encounter during a storm here on Earth (thank goodness), and so the air resistance they hit is significant; the atmosphere, if hit directly, is almost as solid a barrier as encountering rock. Crew-carrying spacecraft will never plunge straight down into the atmosphere, but encounter it at a shallow angle, which allows the craft to encounter the atmosphere’s resistance less abruptly.
So what does this atmospheric resistance do? It slows down the spacecraft, by absorbing some of the spacecraft’s energy. That energy heats up the atmosphere immediately around the craft, encasing the craft in a superheated plasma for part of its descent, until much of the forward motion of the craft has been lost. By approaching the atmosphere at an angle, this process takes a longer time, and the craft can be safely slowed. If we tried to drop straight down into the atmosphere, the craft would not be able to slow down as much, and the sudden increase in pressure from the atmosphere would put so much stress on the craft that it might break. If you have humans in the craft, this is not a good idea. If, on the other hand, you’re just trying to get a satellite out of orbit, you can drop them into the atmosphere at a steeper angle, as they don’t need to be functional when they plunge into the Pacific Ocean. (That’s usually where we put them.)
So yes, there’s a heating problem when you re-enter the atmosphere, but the atmosphere itself isn’t heated any more than ambient air temperature. It's only the air surrounding the craft which heats, and only because there's a spacecraft barreling through. The upper atmosphere is actually quite cold, so there’s no intrinsic heated barrier to traverse. We don’t have the same heating problem when launching a spacecraft, after all. This heating is simply atmospheric drag, though this is dangerous enough - the loss of heat tiles protecting the wings of the space shuttle was what led to the loss of the Shuttle Columbia.
The Van Allen belts, on the other hand, are not actually part of our atmosphere. They’re well beyond it, extending hundreds of miles outwards into space. There are two, both donut-shaped rings surrounding our planet, and are a consequence of our planet’s magnetic field. The Space Shuttle typically orbited at a height of 190 miles to 330 miles above the surface, and the International Space Station orbits at a height of somewhere between 205 and 270 miles above the surface of the Earth.
The innermost Van Allen belt sits somewhere between 400 to 6,000 miles above the surface of our planet. Even if the innermost belt is at its closest, the ISS (and the space shuttle in its day) are more than 100 miles away from the Van Allen Belts. For near-Earth missions, the Van Allen belts are not a hazard to spacefarers.
It was, however, a hazard for the Apollo missions. The Van Allen belts are not a physical barrier to spacecraft, and so, in principle, we could have sent the Apollo spacecraft through the belts. It would not have been a good idea. The Van Allen belts are a kind of trap for charged particles like protons and electrons. They’re held in place by the magnetic field of the Earth, and so they trace the shape of the magnetic field itself. The problem with the Van Allen belts lies not in them being impassable, but in the charged particles they contain.
Charged particles are damaging to human bodies, but the amount of damage done can range from none to lethal, depending on the energy those particles deposit, the density of those particles, and the length of time you spend being exposed to them.
In the case of the Apollo missions, the solution was to minimize the second two factors. We can’t control the energy of those particles, though they can be large. The density of the Van Allen belts is well known (from sending uncrewed probes through them), and there are hotspots you can definitely avoid. In particular, the innermost belt is a rather tightly defined region, and it was possible to stay out of it for the trip to the Moon. The second belt is much larger, and harder to avoid, but there are still denser regions to avoid. For the Apollo trips, we wanted to send the astronauts through a sparse region of the belts, and to try and get through them quickly. This was necessary in any case; the crafts had to make it to the Moon in a reasonable amount of time, and the shorter the trip, the less exposure to all sorts of radiation the astronauts would get.
In the end, it seemed that these tactics worked; the on-board dose counters for the Apollo missions registered average radiation doses to the skin of the astronauts of 0.38 rad. This is about the same radiation dose as getting two CT scans of your head, or half the dose of a single chest CT scan; not too bad, though not something you should do every week.
Your brother is right that both the atmosphere and the Van Allen belts can be dangers to space exploration, but with careful observations, orbital maneuvering, and inventiveness, we’ve navigated our way beyond them many times. Hopefully, we'll continue to do so in the future many times more.
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