I agree with this in principle but in practice specifications seem a quite hard to write and maintain because you have to learn an obscure syntax for a new lang

I agree with this in principle but in practice specifications seem a quite hard to write and maintain because you have to learn an obscure syntax for a new language that nobody on the team knows. So as a devils advocate question: what is the advantage of writing your spec in a spec language vs having it written in a (sufficiently well typed) language and generating the spec from that? Especially when you have one dominant language that most of your code is interfacing with, having the the interface be hand written with idiomatic code for that language is a lot nicer than everyone having to work with auto-generated code that is often not as nice.