Use a separate data structure and then index into it with a linked list like API. No need to do something so dangerous if you simply need an allocation less linked list interface.
But yes there is a place for these data structures. It just shouldn't be common.
If you just mean using array indexes instead of pointers, that can work in some cases, but can run into concurrency issues when the container needs to be resized. With pointers to nodes, new nodes can be allocated and old nodes freed on a separate thread without interfering with the performance sensitive thread, whereas resizing an array requires all access to stop while the backing memory is re-allocated.
I don't think you mean this, but elaborating on why a non-intrusive linked list is not very useful, that's because it takes linear time to find an object in a non-intrusive linked list, but constant time in an intrusive linked list.
For example:
struct Node {
Node* prev_by_last_update;
Node* next_by_last_update;
Node* prev_with_owner;
Node* next_with_owner;
Payload data;
}