The answer to "why not use unsafe" is it's hard to do correctly. What you have to do is pretty well documented https://doc.rust-lang.org/nomicon/working-with-unsafe.html but it's fiddly, tedious, easy to get wrong (you have to understand what the compiler is doing internally), and there are no automated checks to verify you have got it right. A newbie has almost no hope of getting right in the first 5 tries (5 is entirely made up.)
But, Rust already provides a linked list in it's standard library: https://doc.rust-lang.org/std/collections/struct.LinkedList.... So a Rust programmer might say there is no need use unsafe if you want a linked list. There is a "source" link on that page, so it isn't hard to see how it's done. Yes, it does use "unsafe".
Where it gets a little complex is a C programmer will look at that implementation and scoff at the linked list the Rust standard library provides. Amazingly the C and Rust programmers agree that this implementation is mostly useless. We know that because the Rust documentation says: "NOTE: It is almost always better to use Vec or VecDeque because array-based containers are generally faster, more memory efficient, and make better use of CPU cache [than this Rust linked list]."
When a C programmer uses a linked list, he almost invariably uses what someone above termed above an "intrusive" implementation. An intrusive implementation is typically faster, more memory efficient, and makes just as good use of the CPU cache as the Rust Vec or VecDeque. An intrusive implementation reserves a bit of memory inside of the object you want to put in a linked list for the linked list code. That memory is typically "owned" by the linked list code, as in it manages it's lifetime, whereas the lifetime of the object the memory is in is managed by some other mechanism. For example, if you delete some unrelated item it list, it might have to reach in and modify the linked list pointer in this item. But Rust has a simple memory model: if you modify an object you have to prove you own it. The linked list code reaching into another object can not do that.
That's one example, but intrusive linked lists break Rust's ownership and borrowing rules in so many ways, they are like oil and water. As far as I can tell there is no way to provide an intrusive linked list library in Rust that is safe to use, which is to say all code that uses it would have to be flagged as unsafe too.
This probably raises as more questions that it answered, and you are probably wondering how could two implementations of the same thing (a linked list) be so different. I'm not going to go down that rabbit hole, other than to remark it's clear a lot of Rust programmers commenting here don't understand the flexibility the intrusive method gives you. But their instincts are right about one thing: they are dangerous. It's the sort of danger a C programmer is entirely comfortable with, but that same danger is what has lead to C programmers being responsible for a of CVE's. They are right about another thing too: if you put a bit of thought into it, you can usually come up with something that is for all practical purposes just as fast, and the Rust compiler can prove is safe. Sometimes though, "a bit of thought" is an understatement, and sometimes it feels like you are in a straight jacket.
The article code be read is someone straining against that straight jacket, doing a lot of thinking and coming up with what a C programmer would call a fairly ugly (and slower) solution. The old C programmer in me says that's a fair summary. But I also have to acknowledge it is a very safe solution.