I'm not sure what you mean exactly by "tie to the type," though? Just some kind of standard name for it? The benefit of linearity here is that you can change its signature while still getting the compiler to enforce its usage, so you probably wouldn't want it to look like e.g. Rust's Drop trait.

At their core, the definition of linear and affine type systems don't reference SSA or dataflow analysis either. Type systems are generally described using the notation of logic (see this intro if you're unfamiliar: https://langdev.stackexchange.com/a/2693/910), and then to get linearity you define the inference rules such that a use of a resource removes it from the context (one way to write this: https://plato.stanford.edu/entries/logic-linear/#SeqCal).

Things like definite assignment are more user-facing convenience features that are also built on top of the core linear logic.

https://borretti.me/article/how-australs-linear-type-checker...