The question only says that the target list needs to be maintained in sorted order. It doesn't say anything about any other data structure that you may choose to use. The proposed solution first does some preprocessing of the arguments to insert, then does the insertion proper. This is allowed by the problem statement.
A practical reason to do this, rather than insert the elements then sort, would be if the linked list object is shared with another thread that requires it to always be sorted. (In such a scenario, you'd need to ensure that inserting one element is atomic.) So this question isn't just making strange requirements for the sake of being strange. It's the sort of requirements that come up often in the real world of programming.
Another solution with the same complexity would be to insert the elements into the target list as they come, and maintain a parallel data structure mapping element values to node pointers in the target list. To insert each element, find the preceding element in the mapping, and insert the new element after this node. This assumes that the insertion process creates the list nodes as it goes (as opposed to filling existing blank nodes).
This question is more about reading comprehension than about algorithms. The way it's worded, it's a bit of a trick question. It's somewhat poorly worded because it relies on precise reading, but fails to state some key assumptions, such as the fact that obtaining the elements to insert costs $O(n)$, comparing two elements can be done in $O(1)$, and the input domain is effectively unbounded (exercise: come up with an $O(n)$ algorithm if the inputs are integers in the range $[1,42]$). But the given answer is correct.
You made the assumption that there's no way to use an auxiliary data structure. Nothing in the problem statement forbids using auxiliary data structures. A simple way to forbid auxiliary data structures would be to require $O(1)$ memory overhead.
Note that even under this assumption, your reasoning is wrong, or at least imprecise. If you happened to know that the elements are given in the correct order, you could maintain a pointer to the tail of the list, and keep inserting there, which would take $O(n)$. The worst case is not if every element has to be inserted at the last position in the target list, but at the last position reached when traversing the list in some way. The worst case is indeed $\Theta(n^2)$, but to prove this, you have to prove that finding the insertion point in the list takes $\Theta(n)$ time, and this requires proving that the distance from any pointer you have into the list is bounded below by $\Omega(n)$. This is the case if you have a constant number $A$ of pointers (you implicitly assumed $A=1$, with a single pointer at the start of the list), so that you need to traverse at least $k/A$ nodes after $k$ insertions in the worst case.