In what follows, suppose that $|1/0|$ is defined to be equal to $+\infty$.
First, if ${\rm P}(X=0) > 0$, then $|1/X|$ might not even be defined as a random variable, since it is not real-valued. (By a common definition, a random variable is real-valued.)
However, suppose that we allow $|1/X|$ to take the value $+\infty$. Note that $ \bigg|\frac{1}{X}\bigg| \ge \infty {\mathbf 1}(X = 0), $ where ${\mathbf 1}$ is the indicator function. Hence $ {\rm E}\bigg|\frac{1}{X}\bigg| \ge {\rm E}[\infty {\mathbf 1}(X = 0)] = \infty {\rm P}(X = 0). $ So the condition ${\rm E}|1/X| < \infty$ implies that ${\rm P}(X = 0)=0$.
EDIT: With regard to your approach, indeed $ {\rm E}\bigg|\frac{1}{X}\bigg| \ge \int_{\{ \omega :X(\omega ) = 0\} } {\bigg|\frac{1}{{X(\omega )}}\bigg|{\rm P}(d\omega )} . $ Now, using the definition $|1/0|=+\infty$, we have $ \int_{\{ \omega :X(\omega ) = 0\} } {\bigg|\frac{1}{{X(\omega )}}\bigg|{\rm P}(d\omega )} = \int_{\{ \omega :X(\omega ) = 0\} } {\infty {\rm P}(d\omega )} = \infty {\rm P}(X = 0), $ and the conclusion is as before.
Remark: Note that $\infty {\rm P}(X=0)$ is defined to be equal to $0$ when ${\rm P}(X=0)=0$.