Because $$(p_1)(p_2)(p_3)\cdots(p_n)=P(E_1\cap E_2 \cap E_3 \cap \cdots \cap E_n)$$
and the complement of your event space “none of the events occur” is not “all of the events occur,” so you cannot use $P(A')=(p_1)(p_2)(p_3)\cdots(p_n)$ in the formula $P(A)=1-P(A')$.
The actual complement of your event space is “at least one of the events occurs.”
Your event $E$, “none of the events occur,” is equivalent to “$E_1$ does not occur and $E_2$ does not occur and $E_3$ does not occur and . . . and $E_n$ does not occur.”
Using $P(E_k')=1-p_k$, we have
$$\begin{align}
P(E) &= P(E_1' \cap E_2' \cap E_3' \cap \cdots \cap E_n') \\
&= P(E_1')\,P(E_2')\,P(E_3')\cdots P(E_n') \\
&= (1-p_1)(1-p_2)(1-p_3)\cdots(1-p_n) \\
\end{align}$$