Step 3. When $\bar{X}=\bar{x}$ is fixed, $Y_i = \bar{X} - X_i = \bar{x} - X_i$ and since the $X_i$ are independent

The $X_i$ are not independent when you condition on $\bar{X}$.

(and also when you consider to not condition on $\bar{X}$, the $X_i$ are not independent from $\bar{X}$ so the left hand side in $Y_i = \bar{X} - X_i$ does not need to be independent if the $X_i$ are independent)

To solve the problem one might use a geometric trick. The $n$ boundaries for the $Y_i$ are similar to $n-1$ boundaries for the $X_i$ and the solution becomes

$$\mathbb{P}(\max_i|Y_i|\le y_m) = (\Phi(y_m)-\Phi(-y_m))^{n-1}$$

Step 3. When $\bar{X}=\bar{x}$ is fixed, $Y_i = \bar{X} - X_i = \bar{x} - X_i$ and since the $X_i$ are independent

The $X_i$ are not independent when you condition on $\bar{X}$.

(and also when you consider to not condition on $\bar{X}$, the $X_i$ are not independent from $\bar{X}$ so the left hand side in $Y_i = \bar{X} - X_i$ does not need to be independent if the $X_i$ are independent)

Step 3. When $\bar{X}=\bar{x}$ is fixed, $Y_i = \bar{X} - X_i = \bar{x} - X_i$ and since the $X_i$ are independent

The $X_i$ are not independent when you condition on $\bar{X}$.

(and also when you consider to not condition on $\bar{X}$, the $X_i$ are not independent from $\bar{X}$ so the left hand side in $Y_i = \bar{X} - X_i$ does not need to be independent if the $X_i$ are independent)

Step 3. When $\bar{X}=\bar{x}$ is fixed, $Y_i = \bar{X} - X_i = \bar{x} - X_i$ and since the $X_i$ are independent

The $X_i$ are not independent when you condition on $\bar{X}$.

(and also when you consider to not condition on $\bar{X}$, the $X_i$ are not independent from $\bar{X}$ so the left hand side in $Y_i = \bar{X} - X_i$ does not need to be independent if the $X_i$ are independent)

To solve the problem one might use a geometric trick. The $n$ boundaries for the $Y_i$ are similar to $n-1$ boundaries for the $X_i$ and the solution becomes

$$\mathbb{P}(\max_i|Y_i|\le y_m) = (\Phi(y_m)-\Phi(-y_m))^{n-1}$$