Well, my main Elixir project is the backend of a web service. The input is JSON from the frontend and JSON or XML from a number of third party APIs. We process it, hit the db with select, update, insert and usually return a value. The same function with the same inputs usually returns different values because the db is stateful.
What I like of Elixir is not functional programming, it's the extensive use of pattern matching. What I like less is the convoluted syntax of GenServer with all those handle_cast/calls that obfuscate the real code. As a GenServer is usually a way to store a state, they should have had the courage of calling them objects and give them an object like syntax. After all they initially sold Elixir to developers coming from OO languages. If they had given it a Java like syntax maybe it would be 10 or 100 times as big by now.
Thankfully they didn't give it Java like syntax! The syntax is currently a bit more verbose and explicit on dealing with local/micro state which I now like, but without using 100 lines of boilerplate getter/setters. Using pipes gets one enough "OO" syntax feel.
While GenServer's store state they are, in my mind, more akin to microservices than Java/C++ Objects. They're like microservices, but without needing a separate service bus, global naming system (pg2), etc. Now I do wish Dialyzer/Dialyxer also had better support for checking GenServer handler's and messages, especially intra-process. You have a point the syntax their could be spruced up some perhaps. The process / GenServer paradigm kind of remind me of Smalltalk in a way, where you're passing messages that object may or may not want to respond to. That's not really possible with C++/Java objects.
I think you are misunderstanding what GenServers are and are for.
GenServers do have their own state, but they are used for way more than just storing state. It's probably best to literally think of it as a server in your running application, analogous to a server running on a network.
If storing state is all you're trying to do you might want to look at Agent or possibly ETS.
Further, GenServers come from Erlang, not Elixir. They're intended to model all sorts of runtime properties of code.
What I like of Elixir is not functional programming, it's the extensive use of pattern matching. What I like less is the convoluted syntax of GenServer with all those handle_cast/calls that obfuscate the real code. As a GenServer is usually a way to store a state, they should have had the courage of calling them objects and give them an object like syntax. After all they initially sold Elixir to developers coming from OO languages. If they had given it a Java like syntax maybe it would be 10 or 100 times as big by now.