For one thing, the Apollo shield started in the very thin upper atmosphere, and they came in at an angle that meant they bled off as much speed/energy as possible in that thin upper atmosphere before going into the thicker atmosphere.

I don't know that that makes a huge difference to the physics involved, though it certainly may have.

In fact, one of the engineers said that if they came in too steep they'd generate too much heat and probably not survive the re-entry.

But in that case we're talking about human survivability, and a chunk of solid iron is going to survive a whole lot longer than humans or delicate instrumentation. It might look a little worse for the wear, but it's much more likely to be recognizable after the whole experience than anything designed for people.

The layer of air you're talking about at the front of the spacecraft was what heated up the heat shield. Instead of causing heating via friction, the heat was the result of compressing the air.

But after initial heating, the air cushion begins heating itself up instead of the object, reducing the amount of heat the object receives.

The amount of compression you're talking about would be orders of magnitude higher for something starting at 40 km/s in the thick lower atmosphere.

But it would also tail off as the bore cap heated, reducing stresses on it as it went higher.

Also, the Apollo heat shield did heat up to 5000F or 2800C but was designed to be ablative, so that the hot layers burned off and flew off to the sides leaving new material to be heated up and burned off.

True, but on the other hand the a Apollo heat shield wasn't designed to convect heat to other parts of itself. And again, it had a much harder job (keep the Apollo command module at human-survivable temperatures) than the bore cap (not reach the boiling point of iron).

This concrete and metal plug wouldn't have been designed the same way. Concrete apparently melts at 1200C, and steel is approximately the same, so it's very likely some of it melted or vaporized, the question is how much.

All the stuff I read only mentioned the iron, but keep in mind that it has to not only reach the melting point but also undergo phase change, which requires a lot more energy.

I don't know where you're getting the maximum of 22MJ of energy.

11 kJ per m² per second was the peak amount of energy that the Apollo heat shield encountered. Double that for the approximately two seconds it would've been in atmosphere, and it's a pretty handy approximation since the bore cap was about a meter itself.

The whole point of Apollo not going directly into the atmosphere was to take as long as possible to slow down, going through the thinnest part of the atmosphere for as long as possible. [...] One reasonable first approximation of the energy would be to integrate the entire energy per second / power for Apollo's re-entry over the entire 7 minutes (or however long it took until parachutes deployed) and then divide that energy by 2 for the 2 seconds the plug was in the atmosphere.

You're right, the total amount would've been a way better approximation than the peak. Worth looking into.

My guess is that that would have been temperatures well in excess of 1200C which would have made the outer surface start to melt, and most likely a temperature where it just turns to plasma.

I don't have any argument with that. I think the outer surface would definitely have begun to melt.

Would it all have melted / vaporized / plasmafied away? I don't know, it's a huge plug.

Yep. Even just considering the amount of time it would take for the heat to excite all the molecules in the massive chunk of iron, and then for them all to undergo phase change, I just don't think it could've made it.

Since it was launched vertically, anything remaining would probably have come right back down. But, that's assuming it stayed in one piece. I'm guessing it broke apart due to the stresses on it, and breaking apart would have meant more surface area, which would have meant more areas exposed to massive heating, which would have meant more breaking apart.

That's something I couldn't find information on: is iron's tensile strength high enough to prevent the thing shattering apart on contact with air? I'm inclined to think it is—chunks of meteorites bigger than a meter have made it through the atmosphere, for instance. The Hoba meteorite is estimated to only be slightly bigger than it is now before its atmospheric entry, and it's way bigger than the bore cap. Similar composition, too.

TL;DR: I doubt it made it out of the atmosphere.

Either way, I like researching it.

Edit: also, the bore cap starting at the bottom of the atmosphere means that it's likely it experienced less fracture stress, since the air would've accelerated with it rather than being static.