"Maybe it's just due to my problem domain (web development), but it doesn't seem like as big a draw to Elixir and Erlang as pattern matching and FP are."That seems like the perfect domain for this, to me.
HTTP is (traditionally) stateless. So, in the past CGI apps were independent entities. Every new hit was a new world in which the app ran. And, every interaction between each CGI app happened via a standard communication medium (a database or writing to files rather than messages, but if you squint and don't look too closely, you might see the analogy there). There are negatives to that...but, there are real benefits, too. That provided a tremendous level of flexibility in how apps were built and in how they interacted with each other. I'd argue that in some regards (though certainly not all), going to an always-on appserver model was a step backward.
So, to me, in some ways Elixir looks like CGI, only without most of the negatives of CGI. A lot of the mental overhead, like parsing parameters and setting up database connections, can be abstracted away with a modern/fast language designed just for this kind of work. And, the end result is a robust system without a lot of mental overhead other than learning the abstractions and how they interact.
You call out supervisors and message passing and the like as being negatives; complexity you don't want to have to deal with. But, you aren't really comparing apples to apples. Node.js is also asynchronous, and in its bare state is a mess; callbacks are vastly more difficult to interact with and reason about than a system like Erlang/Elixir, and a lot of other languages use the callback model for asynchronous programming.
So, I guess what I'm trying to say is that if you're doing asynchronous programming, Elixir is going to be easier than most (though ES6 has concurrency primitives that are starting to make sense, and Python 3 and Ruby have started getting them, Perl 6 has good options, Perl 5 has somewhat clumsy modules for it, etc.).
All that said, I'm mostly spending my time learning JavaScript/ES6/Node lately; despite its warts, the ecosystem is huge, and there's so many smart people working on the language that by the time of ES7, there will be a very convincing concurrency story. Maybe still not as convincing as Erlang/Elixir and the OTP. But, probably good enough for web apps and services.
I looked into Elixir and Phoenix briefly; watched a few videos, read a few tutorials, never wrote any code. And, came away with a distinct impression of a tiny, tiny, ecosystem. I'm accustomed to going to CPAN or Ruby Gems or PyPI or npm, and finding not just one, but several options for whatever task I want to accomplish, even relatively obscure stuff. It'll be a while before Elixir comes close to even CPAN (which is smaller than all the others these days, though still manages to have modules for nearly every problem I tackle).