What IS the successor function without saying $S(n) = n + 1$?

Yep, that works as a successor function! You do need to make sure the number concept is the minimal concept, otherwise you might have a set that has more than the nodes in the linked-list. Otherwise, the way your doing it is actually pretty swell, and you're right to find the "naive successor function definition" problematic and cyclic.

A successor function $S:\mathbb{N}\to \mathbb{N}$ is any function from which follows the following properties:

• For all $x\in \mathbb{N}, S(x) \neq x$
• $S$ is one-to-one.
• There is some element $e\in\mathbb{N}$ such that, for all $x\in \mathbb{N}, S(x) \neq e$
• Also, $\mathbb{N}$ is the minimal set on which you can define such an $S$.

Any such function and any such set can be one type of successor function. Note, the only things needed for that definition is logic, sets, and functions. I never use addition or zero or any sembalence of numbers. Also, the successor function needn't be unique; multiple different implementations can fit this "successor interface." You can check out a video by PBS on defining the successor function here.