Example 11 Let $n \geqslant 2$ be a positive integer. Find the maximum value of the constant $C(n)$, such that for all real numbers $x_{1}$, $x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in (0,1)(i=1,2, \cdots, n)$, and $\left(1-x_{i}\right)\left(1-x_{j}\right) \geqslant \frac{1}{4}(1 \leqslant i<j \leqslant n)$, we have
$$\sum_{i=1}^{n} x_{i} \geqslant C(n) \sum_{1 \leq i<j \leq n}\left(2 x_{i} x_{j}+\sqrt{x_{i} x_{j}}\right) .$$
First, take $x_{i}=\frac{1}{2}(i=1,2, \cdots, n)$. Substituting into equation (45) gives
$$\frac{n}{2} \geqslant C(n) \mathrm{C}_{n}^{2}\left(\frac{1}{2}+\frac{1}{2}\right)$$

Thus, $C(n) \leqslant \frac{1}{n-1}$.
Next, we prove that $C(n)=\frac{1}{n-1}$ satisfies the condition.
From $1-x_{i}+1-x_{j} \geqslant 2 \sqrt{\left(1-x_{i}\right)\left(1-x_{j}\right)} \geqslant 1(1 \leqslant i<j \leqslant n)$, we get $x_{i}+x_{j} \leqslant 1$.

Taking the sum, we have $(n-1) \sum_{k=1}^{n} x_{k} \leqslant \mathrm{C}_{n}^{2}$, i.e., $\sum_{k=1}^{n} x_{k} \leqslant \frac{n}{2}$.
Therefore,
$$\begin{aligned}
& \frac{1}{n-1} \sum_{1 \leqslant i<j \leqslant n}\left(2 x_{i} x_{j}+\sqrt{x_{i} x_{j}}\right) \\
= & \frac{1}{n-1}\left(2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}+\sum_{1 \leqslant i<j \leqslant n} \sqrt{x_{i} x_{j}}\right) \\
= & \frac{1}{n-1}\left[\left(\sum_{k=1}^{n} x_{k}\right)^{2}-\sum_{k=1}^{n} x_{k}^{2}+\sum_{1 \leqslant i<j \leqslant n} \sqrt{x_{i} x_{j}}\right] \\
\leqslant & \frac{1}{n-1}\left[\left(\sum_{k=1}^{n} x_{k}\right)^{2}-\frac{1}{n}\left(\sum_{k=1}^{n} x_{k}\right)^{2}+\sum_{1 \leqslant i<j \leqslant n} \frac{x_{i}+x_{j}}{2}\right] \\
= & \frac{1}{n-1}\left[\frac{n-1}{n}\left(\sum_{k=1}^{n} x_{k}\right)^{2}+\frac{n-1}{2} \sum_{k=1}^{n} x_{k}\right] \\
= & \frac{1}{n}\left(\sum_{k=1}^{n} x_{k}\right)^{2}+\frac{1}{2} \sum_{k=1}^{n} x_{k} \\
\leqslant & \frac{1}{n}\left(\sum_{k=1}^{n} x_{k}\right) \cdot \frac{n}{2}+\frac{1}{2} \sum_{k=1}^{n} x_{k} \\
= & \sum_{k=1}^{n} x_{k}
\end{aligned}$$

Thus, the original inequality holds.
Therefore, the maximum value of $C(n)$ is $\frac{1}{n-1}$.