Example 5 Given that $a, b, c$ are positive real numbers, satisfying $a+b+c \geqslant a b c$. Prove: at least two of the following three inequalities hold
$$\frac{6}{a}+\frac{3}{b}+\frac{2}{c} \geqslant 2, \frac{6}{b}+\frac{3}{c}+\frac{2}{a} \geqslant 2, \frac{6}{c}+\frac{3}{a}+\frac{2}{b} \geqslant 2 \text {. }$$