What is clear to me is that Rust does incredibly well without it, but wouldn't be hurt by having it. Most of the egregious sins of OOP are mitigated by having far better abstractions available in Rust. ML-style modules, ADTs with pattern matching, type classes, first class functions, etc. Much like scala, if you give them capable FP, OOP stops being abused and starts being used appropriately.
Rust traits and Scala traits (or abstract classes) are more or less equivalent with this one exception. I was just pointed to a proposal that would allow for this in Rust, but it is not yet implemented [0]. Essentially, you would be able to define type members, and then "link" them to a member of the struct/enum that is being "impl"-ed. While ergonomically not quite the same as scala, it is effectively the same as inheriting an abstract class with a constructor parameter.
(The gist was that avoiding mutation and keeping everything purely functional allowed them to roll back and forth in time with no problem, or try out multiple different futures. That's somewhat orthogonal to inheritance.)
[1] https://books.google.co.uk/books?id=ka2VUBqHiWkC&lpg=PA81&ot...
Many C++ game engines also moved away from huge class hierarchies of game objects to property-based game objects that were collections of behavior handlers.