No one is saying assembly language cannot be used to write a dependently typed language system, with a type checker and runtime (which are one and the same in a DT language anyway). However, that does not make assembly language dependently typed.
You are committing the logical error of assuming that the result of a process means inputs to the process are equal to the outputs. Just because I could take grain and make it into bread doesn't mean grain is equivalent to bread.
Assembly language does not have types. What you can do is create a new language with assembly and have that language have types. The reason why it has to be done this way is because these types of checks CANNOT happen at runtime. It's a pre-runtime check that prevents the code from even running if it's not correct.
Think about it. Can you add types to assembly language without creating a new language? No.
Dependent types are not doable in assembly though. Assembly has no notion of types. The claim that all languages are equivalent because they're turing complete has absolutely no relevance here. While true in terms of capability, the syntax and semantics of languages do make them different. One can add a contract library to idris. You cannot do that to assembly code because asembly code as a library. Such a thing is nonsensical since assembly's compilation model has no notion of type checking.