The short “answer” is that the reaction step needs to form a carbon-carbon bond, which isn’t always trivial to make. A fully carbon backbone is likely not reactive in the way they need it to be here.

Certain “weird” metals like zinc, tin, and silicon (And these are the more normal ones) have been found to be able to help with this type of chemistry. The reason for why these metals are useful gets into orbital mechanics, but the core idea is that they can facilitate carbon-carbon bond formation.

The sulphur and fluorine are standard as part of the triflate functional group — it is a stable leaving group based on wanting electrons and being stable once it grabs those electrons (The sulphur can facilitate reasonable stabilisation).

That compound in particular is used to help replace a C-OH bond with a C-CN bond. It may seem like a weird chemical but it's very commonly used in organic synthesis.

> Why wouldn't the same but with a more typical fully-carbon backbone work?

Because a fully carbon backbone would not be as reactive and would not attach/detach from the intermediate as needed