This assumes a very class-oriented type system. Duck-typing and the like don't have the upstream ontology problems.
My main desire from types is as an iteration assistant (with editor integration). Even if I wrote a function myself, I may not remember the exact order of arguments, or the exact name of that one property on the returned map. I want to a) be able to quickly peek and see what those things are - either by visiting the definition or, even better, via a pop-over in my editor - and b) have my editor tell me immediately if I did something dumb so I can correct it and keep moving.
In a dynamic language, whenever I need to double-check the contract for some code, I can't just go look look at its type signature, I have to go read through it. I have to fully load that whole subtree of information into my brain (recursively to any functions it may itself call), when I'm really trying to focus my thoughts on something else. This can be a huge, needless drain on mental resources.
Spec would help with this some, assuming the author follows a good convention of putting all of their assertions at the top of the function. But maybe those assertions are done inside conditionals, creating a more complex type. And maybe my editor doesn't know what to make of them (do any editors? genuinely curious). Etc. It just creates a bunch of little speedbumps to cognition that add up.