If I had guessed at the answer to my own question I would have guessed that the beam has a special ingredient that means that a phoenix server crashes less often than an axum server, programmer skill being equal.
This is sort of the case. Erlang’s design is meant to encourage fault tolerance; what would be an unhandled exception that would crash a process and render a service unreachable in other languages is handled by the BEAM for us. But, the way it does this is not unique. It’s mostly due to an implementation of the actor model baked into the runtime. It can be, and has been, done in other languages, including in Rust. This integration at the core is a point of “more stability” in Elixir/Erlang/Phoenix’s favor, but just because a crash occurs and the runtime handles restarting for you, doesn’t mean that critical sections of your application can’t be down - there is no language or framework that can fix that for you.
I literally don’t know if people loose sleep over this or if they don’t care, or where they land on the spectrum between these two extremes
This is subject to case-by-case. I, like probably many others on this forum and elsewhere, have been woken up by production outages that were “stop the world, this needs fixing now or people can be hurt/sensitive data can be leaked” but sometimes, even at the same organizations/projects an outage is simply "mute this alarm and fix it in the morning. So yes, people do lose sleep over this. You’d probably be interested in looking at software in safety critical systems, e.g., aircraft/medical equipment, to see the types of choices in languages/tools they make. I can tell you Elixir isn’t getting deployed on an airplane, but neither is Axum. 
Edit: You might also enjoy this video about Zig, or the first few minutes at least.
Edit 2: One more thing on the “do people lose sleep over this,” in cases where a software defect is worth sleeping over I wouldn’t say between Elixir/Axum one is more or less better/worse than the other. The first thing I’d evaluate on making a choice is the team and the domain.
If I understand you correctly, both technologies provide programmers more or less the same overall level of safety against servers crashing, and it’s really all about how the programmers use those tools… 
