Doesn't seem like it necessarily stops inheritance. You need to throw away method references or you're just back to open faced classes.
You can still write new methods that blow away any assumptions previously made about type state,causing bugs. It seems like it could be even worse seeing as you possibly have less access controls for fields.
With the typing system of Haskell, maybe Rust too, you can do a lot to prevent the type from being able to represent wrong state. Immutability also does a lot to prevent issues with false state and public fields.
If you want to prevent people outside the module from adding functions, you can do that in Haskell by only exporting the type-name. This prevents other modules from actually accessing any data inside the type, whilst still allowing them to use the type in functions. This includes sometimes making getter functions to read values from the type. I'm pretty sure Rust allows for something similar.
In general, FP still allows for encapsulation. Moreover, it uses immutability and to some extend the type-system to prevent illegal states of data types.
No, a class is a data type coupled to functions operating on that class. For the purpose of this discussion a module is a way to semantically group types and functions without coupling them (languages like OCaml get fancy by actually typing modules which enables some neat ideas, but is not important to this discussion).