I think you are confusing independent events with disjoint events.
Independent events are defined by $P(A\cap B) = P(A)P(B)$. $A$ and $B$ will overlap in the Venn diagram, except in the case of $P(A) = 0$ or $P(B) = 0$.
Disjoint events are given by $P(A\cap B) = 0$, meaning it is impossible for the events to occur together. Here the Venn diagram areas have no overlap.
Venn diagrams are not a formal proof, nor a substitute for it, just an illustrative tool that can be useful as a guiding tool for your narrative/proof.
If writing a formal proof for this law, you will need to show
$$A \cup (B \cap C) \subseteq (A \cup B) \cap (A \cup C) \;\;\; \text{and} \;\;\; (A \cup B) \cap (A \cup C) \subseteq A \cup (B \cap C)$$
and then use the fact that if $X \subseteq Y$ and $Y \subseteq X$, then $X = Y$.
If you don't need formality, then in the appropriate context it can be used, I suppose. And, up to your ability to produce said diagrams, you could use a Venn diagram of $n$ circles, depending on what you're proving, but it gets messy quick so I wouldn't recommend it for more than $3$ sets.
In short, it depends on the level of formality that is expected of you. There's no denying that Venn diagrams in contexts like these are super, super helpful in illustrating concepts, and can be taken as a sort of heuristic proof, but they are not a substitute for formal proofs.
I say this in light of the assumption that you are probably encountering this in a class of some sort like a number of questions here. Classes in set theory, generally, will expect formality, not Venn diagrams, for example. In research, publications, journals, etc., things are much, much murkier depending on the context.
Best Answer
The problem seems to be that the formulation "occurs at least one not" could possibly be said to be ambiguous.
Under a very strange reading of the assignment, one could interpret the phrase "at least one not" as "it is not the case that at least one of the events occurs", $\overline{A \cup B \cup C}$, which is equivalent to "none of the events occurs", $\overline{A} \cap \overline{B} \cap \overline{C}$, and thus corresponds to the Venn diagram suggested in the solution.
However, the more straightforward (and in my judgement, the only acceptable from a grammatical point of view) reading is that "for at least one of the events it holds that it does not occur", $\overline{A} \cup \overline{B} \cup \overline{C}$, equivalently "it is not the case that all of the events occur", $\overline{A \cap B \cap C}$, which corresponds to the situation you correctly pictured in your Venn diagram.
I find it very hard to see that the first reading would be the expected interpretation or even possible at all, and think that your professor should have accepted your solution.