Meaning of the Time Hierarchy Theorem
You miss the meaning of the Time Hierarchy theorem, or perhaps of the classes TIME($n^k$) that you show.

It is indeed obvious that $O(n^k) \subsetneq O(n^{k+1})$, for instance. It is also clear that the set of Turing machines that run in time $O(n^k)$ is a strict subset of those that run in $O(n^{k+1})$.

However, it is not immediately obvious that the set of problems that can be solved by TMs in time $O(n^k)$ (i.e., the class TIME($n^k$)) is a strict subset of TIME($n^{k+1})$. For this much stronger statement, you need the Time Hierarchy Theorem. That is, you need to argue that there exists some problem that can be solved in time $n^{k+1}$ but not in time $n^k$. (Of course, this is just a special case of applying the theorem to polynomial running times.)